<PAGE>   1

                                                                  Exhibit (4)(i)


                    THIRD AMENDMENT TO LIMITED TERM EXTENSION


                  This Third Amendment to Limited Term Extension (the "Third
Amendment") is entered into as of this 29th day of May, 1998 between National
Auto Credit, Inc. ("National"), as borrower, NAC, Inc. and NAC Investment
Company, as guarantors, The First National Bank of Chicago ("First Chicago"),
Morgan Guaranty Trust Company of New York ("Morgan"), The Bank of New York
("BNY"), First Union National Bank ("First Union"), The Huntington National Bank
("Huntington"), Allstate Life Insurance Company ("Allstate"), Connecticut
General Life Insurance Company (on behalf of itself and one or more separate
accounts, collectively "Connecticut General"), Principal Mutual Life Insurance
Company ("Principal Mutual"), New York Life Insurance Company ("New York Life"),
New York Life Insurance and Annuity Corporation ("New York Annuity"), Lincoln
National Life Insurance Company ("Lincoln National") and Lincoln Life & Annuity
Company of New York ("Lincoln Life"), as lenders, and NBD Bank, ("NBD") as
issuer of letters of credit. Capitalized terms not otherwise defined herein
shall have the meanings ascribed to such terms in that certain Limited Term
Extension dated as of February 13, 1998 (as amended to the date hereof, the
"Extension").

                                    RECITALS
                                    --------

                  WHEREAS, National presently owes (i) approximately $29.7
million in Principal Obligations on Loans extended by the Banks pursuant to the
Credit Agreement, and (ii) approximately $35.1 million in Principal Obligations
on loans extended by the Insurance Companies pursuant to the terms of the Note
Purchase Agreement and Notes.

                  WHEREAS, on February 13, 1998, National, the Lenders and NBD
entered into the Extension pursuant to which the Banks granted National a
limited extension of maturities and the Insurance Companies granted National a
limited extension of the Waiver Extension, all on the terms and conditions
specifically set forth therein.

                  WHEREAS, pursuant to the terms of the Extension, all Lender
Obligations became due and payable in full on March 6, 1998.

                  WHEREAS, on March 6, 1998 and March 16, 1998, National, the
Lenders and NBD entered into the First Amendment to Limited Term Extension (the
"First Amendment") and the Second Amendment to Limited Term Extension (the
"Second Amendment"), respectively, pursuant to which the Lenders granted
National additional limited extensions of maturities and the Insurance Companies
granted National additional limited extensions of the Waiver Extension, all on
the terms and conditions specifically set forth therein.

<PAGE>   2


                  WHEREAS, National has requested the Lenders and NBD for an
additional extension of the Extension Termination Date.

                  WHEREAS, the Lenders and NBD are prepared to extend the
Extension Termination Date until September 30, 1998 on the terms and conditions
contained herein.

                  NOW, THEREFORE, in consideration of the foregoing and for
other good and valuable consideration, the receipt and sufficiency of which are
hereby acknowledged, the parties hereto agree to amend the Extension as follows:

                  1. The definition of "Extension Termination Date" is amended
in its entirety so that, as amended, it provides as follows:

         "EXTENSION TERMINATION DATE" means the earlier to occur of (i)
         September 30, 1998, and (ii) the occurrence of an Extension Default.

                  2. The following definition is added:

         "PRIME RATE" means the rate of interest announced from time to time by
the Agent as its prime rate.

                  3. Paragraph 3 of the Extension is amended in its entirety so
that, as amended, it provides as follows:

                  "3. From January 31, 1998 through May 29, 1998, interest will
         accrue on the unpaid balance of all Lender Obligations at a per annum
         rate (computed on the basis of a 360-day year of twelve 30-day months)
         equal to the higher of (i) the Floating Rate, and (ii) 7.66%. After May
         29, 1998, interest will accrue on the unpaid balance of all Lender
         Obligations at a per annum rate (computed on the basis of a 360-day
         year of twelve 30-day months) equal to the higher of (i) the Prime Rate
         plus 2%, and (ii) 9.66%. Commencing on March 20, 1998 interest will be
         payable to all Lenders weekly, in arrears, on Friday of each week. From
         and after May 29, 1998 interest will accrue on the unpaid balance of
         any overdue Credit Agreement Obligation or Note Obligation at a per
         annum rate (computed on the basis of a 360-day year of twelve 30-day
         months) equal to the Prime Rate plus 4%."

                  4. Paragraph 4 of the Extension is amended in its entirety so
that, as amended, it provides as follows:

                  "4. Subject to the provisions of paragraph 17 below, National
         shall make principal payments to the Lenders and NBD on the following
         dates and in the following amounts:


<PAGE>   3


                             February 19, 1998                $1,000,000
                             March 6, 1998                    $5,500,000
                             March 16, 1998                   $6,000,000
                             May 29, 1998                     $9,000,000

         Additionally, National shall make payments to the Lenders and NBD on
         the dates and in the amounts set forth on Exhibit A, to be applied
         first to accrued and unpaid interest, and the balance to principal.
         Until all Lender Obligations and the Reimbursement Obligation have been
         paid in full, National shall also remit to the Lenders all tax refunds
         received by National (except as otherwise provided in the security
         agreement to be executed by the parties), when and as such refunds are
         received. Each principal payment (and such tax refunds) will be
         distributed to the Lenders and NBD on a Pro Rata basis and will be
         applied by the Lenders, except as provided in paragraph 17 below,
         against the Principal Obligations. Principal payments distributed to
         NBD in respect of the Reimbursement Obligation will be held by NBD as
         cash collateral to secure payment of the Reimbursement Obligation until
         the Letters of Credit are drawn. If one or more Letters of Credit
         expires or is returned to NBD undrawn, NBD will distribute the cash
         collateral then held by NBD in respect of such Letter(s) of Credit to
         all Lenders (and, to the extent that other Letters of Credit remain
         outstanding, to NBD) for reduction of Principal Obligations on a Pro
         Rata basis. The remaining balance of all Lender Obligations shall be
         due and payable in full on the Extension Termination Date; provided,
         however, that to the extent payment of the Note Obligations is
         voluntarily waived or extended by the Insurance Companies beyond the
         Extension Termination Date, the Extension Termination Date shall not
         constitute a "Settlement Date" for purposes of Section 8.6 of the Note
         Purchase Agreement."

                  5. Paragraph 9 of the Extension is amended in its entirety so
that, as amended, it provides as follows:

                  "9. Notwithstanding any prior or future waiver of defaults by
         the Lenders (and without waiving or modifying rights available to the
         Lenders under the Credit Agreement and the Note Purchase Agreement),
         (i) National will, on a monthly basis or such shorter time as the
         Lenders may request, promptly pay statements for the reasonable fees
         and expenses incurred on behalf of the Lenders by the firms of Policano
         & Manzo, LLC, O'Melveny & Myers LLP, and Sidley & Austin, and (ii)
         National will, on a monthly basis, reimburse each Lender for its
         reasonable fees and expenses (including reasonable fees and expenses
         for outside and in-house counsel as well as travel and other expenses
         for each Lender's personnel) incurred by such Lender in connection with
         such Lender's loans to National in accordance with the terms of the
         Credit Agreement and the Note Purchase Agreement, as the case may be.
         Additionally, it is an express condition to the effectiveness of this
         Amendment that all outstanding statements for fees and expenses
         incurred by Policano & Manzo, LLC, O'Melveny & Myers LLP, and Sidley &
         Austin shall have been paid in full in connection with the matters set
         forth herein. The Lenders will provide National with


<PAGE>   4


         copies of the billing information submitted to the Lenders by Policano
         & Manzo, LLC, O'Melveny & Myers LLP, and Sidley & Austin; provided,
         however, that the Lenders shall be entitled to redact information which
         the Lenders believe to be subject to privilege or to be otherwise
         confidential."

                  6. The following paragraphs shall be added to the Extension as
paragraphs 16 and 17, respectively:

                  "16. National, NAC, Inc and NAC Investment Company shall each
         execute (and shall cause their affiliate, ARAC, Inc. ("ARAC") to
         execute) security agreements substantially in the form of attached
         Exhibit A and shall thereby each grant to the Lenders and NBD a
         security interest in and lien upon all of their respective right, title
         and interest in and to federal and state income tax refunds to secure
         payment of the Reimbursement Obligation and all Lender Obligations, as
         well as NAC, Inc.'s and NAC Investment Company's guaranties of the
         Lender Obligations."

                  "17. If NAC fails to pay the Lender Obligations in full on or
         before September 30, 1998, the Lenders shall be entitled to receive
         from NAC, on September 30, 1998, an amendment fee in the amount of
         $500,000. In such event, (i) the amendment fee shall be deemed to have
         been paid with $500,000 of the monies paid to the Lenders on May 29,
         1998, (ii) $500,000 of Principal Obligations previously paid with such
         sum shall automatically reinstate, effective as of May 29, 1998, and
         (iii) NAC shall immediately pay to the Lenders interest accrued on the
         reinstated principal from May 29, 1998 through the date of payment at
         the rate specified in paragraph 3 above."

                  7. Except to the extent that they are amended by the terms of
this Third Amendment, the terms and provisions of the Extension shall remain in
full force and effect.

                  8. This Third Amendment is a contract made under, and shall be
governed by and construed in accordance with, the law of the State of Illinois
applicable to contracts made and to be performed entirely within such State and
without giving effect to choice of law principles of such State.

                                       NATIONAL AUTO CREDIT, INC.

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------

                                       NAC, INC.

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


<PAGE>   5


                                       NAC INVESTMENT COMPANY

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       THE FIRST NATIONAL BANK OF CHICAGO, as
                                       administrative Agent and as a Lender

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       MORGAN GUARANTY TRUST COMPANY OF
                                       NEW YORK,  as Documentation Agent and as
                                       a Lender

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       THE BANK OF NEW YORK

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       FIRST UNION NATIONAL BANK (formerly known
                                       as First Union National Bank of North
                                       Carolina)

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------




                                       THE HUNTINGTON NATIONAL BANK

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


<PAGE>   6


                                       ALLSTATE LIFE INSURANCE COMPANY

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       CONNECTICUT GENERAL LIFE INSURANCE
                                       COMPANY
                                       BY CIGNA INVESTMENTS, INC.

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       CONNECTICUT GENERAL LIFE INSURANCE
                                       COMPANY ON BEHALF OF ONE OR MORE SEPARATE
                                       ACCOUNTS BY CIGNA INVESTMENTS, INC.

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       PRINCIPAL MUTUAL LIFE INSURANCE COMPANY

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------



                                       NEW YORK LIFE INSURANCE COMPANY

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------

<PAGE>   7


                                       NEW YORK LIFE INSURANCE AND ANNUITY
                                       CORPORATION BY NEW YORK LIFE INSURANCE
                                       COMPANY

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       LINCOLN NATIONAL LIFE INSURANCE COMPANY

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       LINCOLN LIFE & ANNUITY COMPANY OF
                                       NEW YORK

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------


                                       NBD BANK

                                       By:
                                           ----------------------------------
                                       Its
                                          -----------------------------------



<PAGE>   8



                                    EXHIBIT A

                      SCHEDULE OF REQUIRED WEEKLY PAYMENTS


<TABLE>
<S>                                                           <C>
                           March 20, 1998                     $625,000
                           March 27, 1998                     $625,000
                           April 3, 1998                      $625,000
                           April 10, 1998                     $625,000
                           April 17, 1998                     $625,000
                           April 24, 1998                     $625,000
                           May 1, 1998                        $625,000
                           May 8, 1998                        $625,000
                           May 15, 1998                       $625,000
                           May 22, 1998                       $625,000
                           May 29, 1998                       $625,000
                           June 5, 1998                      $1,000,000
                           June 12, 1998                     $1,000,000
                           June 19, 1998                     $1,000,000
                           June 26, 1998                     $1,000,000
                           July 3, 1998                      $1,000,000
                           July 10, 1998                     $1,000,000
                           July 17, 1998                     $1,000,000
                           July 24, 1998                     $1,000,000
                           July 31, 1998                     $1,000,000
                           August 7, 1998                    $1,000,000
                           August 14, 1998                   $1,000,000
                           August 21, 1998                   $1,000,000
                           August 28, 1998                   $1,000,000
                           September 4, 1998                 $1,000,000
                           September 11, 1998                $1,000,000
                           September 18, 1998                $1,000,000
                           September 25, 1998                $1,000,000
</TABLE>