<PAGE>

                                                                   Exhibit 10(e)


                                FIRST AMENDMENT

     THIS FIRST AMENDMENT dated as of August 12, 1999 (this "Amendment") is to
                                                             ---------
the Term Loan Agreement (the "Term Loan Agreement") dated as of July 15, 1999
                              -------------------
among UNITED RENTALS (NORTH AMERICA), INC. (the "Company"), UNITED RENTALS, INC.
                                                 -------
("Parent"), various financial institutions and BANK OF AMERICA, N.A. (f/k/a Bank
  ------
of America National Trust and Savings Association), as Administrative Agent (in
such capacity, the "Agent").  Unless otherwise defined herein, capitalized terms
                    -----
used herein have the respective meanings set forth in the Term Loan Agreement.


     WHEREAS, the Company, Parent, the Agent, Bank of America, N.A. ("BofA") and
                                                                      ----
Goldman Sachs Credit Partners L.P. ("Goldman Sachs") have entered into the Term
                                     -------------
Loan Agreement; and

     WHEREAS, the parties hereto desire to amend the Term Loan Agreement to (a)
add the parties listed on the signatures hereof under the heading "New Lenders"
(collectively the "New Lenders") as "Lenders" thereunder and (b) make certain
                   -----------
other changes as hereinafter set forth;

     NOW, THEREFORE, in consideration of the premises and for other good and
valuable consideration (the receipt and sufficiency of which are hereby
acknowledged), the parties hereto agree as follows:

     SECTION 1 AMENDMENT.  Effective on (and subject to the occurrence of) the
               ---------
First Amendment Effective Date (as defined below), the Term Loan Agreement shall
be amended as set forth below.

     1.1  Section 14.9.1.  Section 14.9.1 is amended by (a) inserting the
          ---------------
following language after the word "withheld" at the end of the first
parenthetical clause therein:  "and shall not be required for any assignment and
delegation to (a) another Lender or (B) a Related Fund" and (b) deleting the
second parenthetical therein (which begins with the language "(provided that no
                                                               --------
written consent").

     1.2  Schedule 1.1(A)  Schedule 1.1(A) is amended in its entirety by
          ---------------
substituting the Schedule 1.1(A) attached hereto therefor.
                 ---------------
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     1.3  Schedule 14.3.  Schedule 14.3 is amended in its entirety by
          -------------
substituting the Schedule 14.3 attached hereto therefor.
                 -------------


     SECTION 2 REPRESENTATIONS AND WARRANTIES. Each of Parent and the Company
               ------------------------------
represents and warrants to the Agent and the Lenders that (a) the
representations and warranties made in Section 9 of the Term Loan Agreement are
true and correct on and as of the First Amendment Effective Date with the same
effect as if made on and as of the First Amendment Effective Date (except to the
extent such representations and warranties expressly refer to an earlier date,
in which case they were true and correct as of such earlier date); (b) no Event
of Default or Unmatured Event of Default exists or will result from the
execution of this Amendment; (c) no event or circumstance has occurred since the
Closing Date that has resulted, or would reasonably be expected to result, in a
Material Adverse Effect; (d) the execution and delivery by Parent and the
Company of this Amendment, the execution and delivery by the Company of the New
Notes (as defined below), the performance by Parent and the Company of their
respective obligations under the Term Loan Agreement as amended hereby (as so
amended, the "Amended Agreement") and the performance by the Company of the New
              -----------------
Notes (i) are within the corporate powers of Parent and the Company, (ii) have
been duly authorized by all necessary corporate action on the part of Parent and
the Company, (iii) have received all necessary governmental approval and (iv) do
not and will not contravene or conflict with any provision of law or of the
charter or by-laws of Parent or the Company or of any indenture, loan agreement
or other contract, or any order or decree, which is binding upon Parent or the
Company; and (e) each of the Amended Agreement and each New Note is the legal,
valid and binding obligation of Parent and the Company, as applicable,
enforceable against Parent and the Company, as applicable, in accordance with
its terms, except as enforceability may be limited by applicable bankruptcy,
insolvency, or similar laws affecting the enforcement of creditors' rights
generally or by equitable principles relating to enforceability.

     SECTION 3 EFFECTIVENESS.  The amendments set forth in Section 1 above shall
               -------------                               ---------
become effective, as of the day and year first above written, on such date (the
"First Amendment Effective Date") when the Agent shall have received, (a) a
 ------------------------------
counterpart of this Amendment executed by each of the parties

                                      -2-
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hereto (or, in the case of any party from which the Agent has not received a
counterpart hereof, facsimile confirmation of the execution of a counterpart
hereof by such party) and (b) each of the following documents, each in form and
substance satisfactory to the Agent:

      3.1  Notes.  New Notes, substantially in the form of Exhibit A to the Term
           -----
Loan Agreement, payable to the order of each of the Lenders (collectively, the

"New Notes").
----------


      3.2  Other Documents.  Such other documents as the Agent or any Lender may
           ---------------
reasonably request in connection with Parent's and the Company's authorization,
execution and delivery of this Amendment.

     SECTION 4  ADDITION OF LENDERS.  On the First Amendment Effective Date,
                -------------------
each New Lender shall become a "Lender" under and for all purposes of the
Amended Agreement, shall be bound by the Amended Agreement, and shall be
entitled to the benefits of the Amended Agreement and each other Loan Document,
and each Lender (including BofA and Goldman Sachs) shall have a Term Loan in the
amount, and a Percentage, as set forth on Schedule 1.1(A) hereto.  To facilitate
                                          ---------------
the foregoing, each New Lender agrees that on the First Amendment Effective
Date, it will remit to the Agent funds in an amount equal to its Term Loan, and
the Agent agrees to immediately (i) remit a portion of such funds to BofA and
Goldman Sachs, in such amounts as are necessary to reduce the Term Loans of BofA
and Goldman Sachs to the amounts set forth opposite their names on Schedule
                                                                   --------
1.1(A) hereto and (ii) to remit the balance of such funds to the Company.  Each
------
New Lender agrees that all interest and fees accrued under the Term Loan
Agreement prior to the First Amendment Effective Date are the property of BofA
and Goldman Sachs.  By their signatures below each of BofA and Goldman Sachs
confirms that it has not sold or otherwise encumbered its rights under the Term
Loan Agreement or its interest in any Loans prior to the syndication thereof
pursuant to this Amendment.

     SECTION 5  MISCELLANEOUS.
                -------------

     5.1  Continuing Effectiveness, etc.  As herein amended, the Term Loan
          ------------------------------
Agreement shall remain in full force and effect and is hereby ratified and
confirmed in all respects.  After the First Amendment Effective Date, all
references in the Term

                                      -3-
<PAGE>

Loan Agreement, the Notes, each other Loan Document and any similar document to
the "Term Loan Agreement" or similar terms shall refer to the Amended Agreement.

     5.2  Counterparts.  This Amendment may be executed in any number of
          ------------
counterparts and by the different parties on separate counterparts, and each
such counterpart shall be deemed to be an original but all such counterparts
shall together constitute one and the same Amendment.

     5.3  Expenses.  The Company agrees to pay the reasonable costs and expenses
          --------
of the Agent (including attorney costs) in connection with the preparation,
execution and delivery of this Amendment.


     5.4  Governing Law.  This Amendment shall be a contract made under and
          -------------
governed by the internal laws of the State of Illinois.

     5.5  Successors and Assigns.  This Amendment shall be binding upon Parent,
          ----------------------
the Company, the Lenders and the Agent and their respective successors and
assigns, and shall inure to the benefit of Parent, the Company, the Lenders and
the Agent and the successors and assigns of the Lenders and the Agent.

                                      -4-
<PAGE>

     Delivered at New York, New York, as of the day and year first above
written.

                             UNITED RENTALS, INC.


                             By__________________________
                               Title_____________________



                             UNITED RENTALS (NORTH AMERICA), INC.


                             By__________________________
                               Title_____________________



                             BANK OF AMERICA, N.A., as
                             Administrative Agent


                             By__________________________
                               Title_____________________



                             BANK OF AMERICA, N.A., as a Lender


                             By_________________________
                               Title____________________



                             GOLDMAN SACHS CREDIT PARTNERS,
                             L.P., as Syndication Agent and as
                             a Lender

                                      -5-
<PAGE>

                             By_________________________
                               Title____________________



                             "NEW LENDERS":
                              -----------


                             DEUTSCHE BANK AG, New York and/or
                               Cayman Islands Branch


                             By_________________________
                               Title____________________




                             NORTH AMERICAN SENIOR FLOATING
                             RATE FUND
                             By:CypressTree Investment Company,
                             Inc., as Portfolio Manager


                             By_________________________
                               Title____________________



                             CYPRESSTREE SENIOR FLOATING RATE
                             FUND
                             By:
                             CypressTree Investment Management
                             Company, Inc., as Portfolio Manager


                             By_________________________
                               Title____________________


                                      -6-
<PAGE>

                             KZH CYPRESSTREE - 1 LLC


                             By_________________________
                               Title____________________


                                      -7-
<PAGE>

                             CYPRESSTREE INSTITUTIONAL FUND,
                             LLC
                             By:CypressTree Investment Management
                             Company, Inc., as Portfolio
                             Manager


                             By_________________________
                               Title____________________



                             FLOATING RATE PORTFOLIO


                             By_________________________
                               Title____________________



                             GENERAL ELECTRIC CAPITAL CORPORATION

                             By_________________________
                               Title____________________



                             HARRIS TRUST AND SAVINGS BANK


                             By_________________________
                               Title____________________


                                      -8-
<PAGE>

                             IKB DEUTSCHE INDUSTRIEBANK AG
                             LUXEMBOURG BRANCH


                             By_________________________
                               Title____________________



                             METROPOLITAN LIFE INSURANCE
                             COMPANY


                             By_________________________
                               Title____________________



                             MONY LIFE INSURANCE COMPANY


                             By_________________________
                               Title____________________



                             PARIBAS CAPITAL FUNDING LLC


                             By_________________________
                               Title____________________


                                      -9-
<PAGE>

                             JACKSON NATIONAL LIFE INSURANCE
                             COMPANY


                             By_________________________
                               Title____________________



                             TYLER TRADING, INC.


                             By_________________________
                               Title____________________



                             THE PRUDENTIAL INSURANCE COMPANY
                             OF AMERICA


                             By_________________________
                               Title____________________



                             UNION BANK OF CALIFORNIA, N.A.


                             By_________________________
                               Title____________________


                                     -10-
<PAGE>

                             COMERICA BANK


                             By_________________________
                               Title____________________



                             BANK OF NOVA SCOTIA


                             By_________________________
                               Title____________________



                             THE FUJI BANK, LIMITED


                             By_________________________
                               Title____________________



                             HELLER FINANCIAL INC.


                             By_________________________
                               Title____________________


                                     -11-
<PAGE>

                             FREMONT INVESTMENT AND LOAN


                             By_________________________
                               Title____________________



                             CONTINENTAL ASSURANCE COMPANY
                             Separate Account (E)
                             By:
                             TCW Asset Management Company as
                             Attorney-in-Fact


                             By_________________________
                               Title____________________


                             By_________________________
                               Title____________________



                             UNITED OF OMAHA LIFE INSURANCE
                             COMPANY
                             By:TCW Asset Management Company, its
                             Investment Advisor


                             By_________________________
                               Title____________________


                             By_________________________
                               Title____________________

                                     -12-
<PAGE>

                             SEQUILS I, LTD.


                             By_________________________
                               Title____________________


                             By_________________________
                               Title____________________



                             KZH CRESCENT-2 LLC


                             By_________________________
                               Title____________________



                             KZH CRESCENT LLC


                             By_________________________
                               Title____________________


                                     -13-