Limit

x

{\displaystyle x}

sin
⁡
x

x

{\displaystyle \frac{\sin x}{x}}
 0.01 | 0.999983...

f
:
(
a
,
b
)
→

R

{\displaystyle f\colon (a,b) \to \R}

 function on open interval that is the set of the set of real numbers and < math>p, L \in\R</math> is a real number where

p
∈
(
a
,
b
)

{\displaystyle p \in (a,b )}

 resist**limit of

f

{\displaystyle f}

when

x

{\displaystyle x}

 tends to be bad.

p

{\displaystyle p}

L

{\displaystyle L}** only if

Remember the value

ε
>
0

{\displaystyle \varepsilon > 0}

 \vert x - p \right\vert < \delta</math> if

f
(
x
)
−
L

<
ε
.

{\displaystyle \left\vert f(x) - L \right\vert < \varepsilon . }

limit cutting machine

f

{\displaystyle f}

lim

x
→
p

f
(
x
# )
L

{\displaystyle \lim\_{x \to p} f(x) = L}

or one can be

f
(
x
)
→
L

{\displaystyle f(x) \to L}

 when

x
→
p

{\displaystyle x \to p}

 (read "

f
(
x
)

{\displaystyle f(x)}

 is likely **L

{\displaystyle L}** when

x

{\displaystyle x}

 tends to be low

p

{\displaystyle p}

")

Limit definitions are often not appropriate for the value of

f

{\displaystyle f}

 where

p

{\displaystyle p}

 is appropriate,

f
(
p
)

{\displaystyle f(p) }

 does not need to be evaluated.

## PRINCE PRINCIPAL

Nose *f* : (*M*,d*M*) -> (*N*,d' ' N*) is the target value between (as a function defined on) jet space two spaces, and can do* p *∈*M *and* L  *∈*N*, the cleanliness "limit of* f *where* p *is* L*" and writes:

lim

x
→
p

f
(
x
# )
L

{\textstyle \lim\_ {x \to p}f(x) = L }

only if stores the value of ε > 0 at δ > 0 where* stored'. ' *x* ∈*M* *and* d*M*(*x*, *p* ) < δ *Then*, d*N*(*f*(*x*), *L*) < ε

#### The limit of the actual value target at a particular point

Let *f* be true.

lim

x
→
p

f
(
x
# )
L

{\displaystyle \lim\_{x \to p} f(x) = L}

or we will write

lim

x
→
p

f
(
x
# )
∞

{\displaystyle \lim\_{x \to p}f(x) = \infty}

 only if
Threaten the value of *R* > 0 (big pepper) to do δ > 0 coordinates 1 value at *real number coordinates* *x* *at* 0 < |*x* -*p*| < δ, *f* (*x*) > *R*;

or to write

lim

x
→
p

f
(
x
# )
−
∞

{\displaystyle \lim\_{x \to p}f(x) = -\infty}

 only if
slant the value of *R* < 0 in the world δ > 0 coordinates 1 value at *real number coordinates* *x* *at* 0 < |*x*- *p*| < δ, *f* (*x*) < *R*

Razor in definition, instead of *right limit* is written by :

lim

x
→

p

{\displaystyle \lim\_{x \to p^+}}

 and if *p*-*x* can be used instead ' 'Left limit *replaced by :

lim

x
→

p
−

{\displaystyle \lim\_{x \to p^-}}*

#### Limit of real value at infinity

Let *f*(*x*) queue actual expected limit of when *x* is suitable to...

writing posture

lim

x
→
∞

f
(
x
# )
L

{\displaystyle \lim\_{x \to \infty}f(x) = L}

### Complex

complex plane with metric as

d
(
x
,
y
)
:=

x
−
y

{\displaystyle d(x, y) := |x-y|}

 will be metric space ...
Two visions are spreading towards each other.

### example limit

lim

x
→
3

x
2

=
9

{\displaystyle \lim\_{x \to 3}x^2=9}

lim

x
→

0

x
x

=
1

{\displaystyle \lim\_{x \to 0^+}x^x=1}

lim

x
→
0

1
x

=

Undefined

{\displaystyle \lim\_{x \to 0}{1 \over x} = \mbox{Undefined}}

lim

x
→

0

1
x

=
∞

{\displaystyle \lim\_{x \to 0^+}{1 \over x} = +\infty}

limit of 1/*x* when *x* approaching 0 from the right is +∞

lim

x
→

0

x

x

=
1

{\displaystyle \lim\_{x \to 0^+}{|x| \over x}=1}

lim

x
→

0
−

x

x

=
−
1

{\displaystyle \lim\_{x \to 0^-}{|x| \over x}=-1}

lim

x
→
0

x
sin
⁡

1
x

=
1

{\displaystyle \lim\_{x \to 0}x \sin {1 \over x} = 1}

from:

1. ↑ no reference