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(e^x-e)/(-1+x)
Limit of the function/ (e^x-e)/(-1+x)

Limit of the function (e^x-e)/(-1+x)

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     / x    \
     |E  - E|
 lim |------|
x->1+\-1 + x/
lim⁡x→1+(ex−ex−1)\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right) 
Limit((E^x - E)/(-1 + x), x, 1)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→1+(ex−e)=0\lim_{x \to 1^+}\left(e^{x} - e\right) = 0
and limit for the denominator is
lim⁡x→1+(x−1)=0\lim_{x \to 1^+}\left(x - 1\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→1+(ex−ex−1)\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)
=
Let's transform the function under the limit a few
lim⁡x→1+(ex−ex−1)\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)
=
lim⁡x→1+(ddx(ex−e)ddx(x−1))\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(e^{x} - e\right)}{\frac{d}{d x} \left(x - 1\right)}\right)
=
lim⁡x→1+ex\lim_{x \to 1^+} e^{x}
=
lim⁡x→1+ex\lim_{x \to 1^+} e^{x}
=
ee
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
-2.0-1.5-1.0-0.52.00.00.51.01.505
Plot the graph
One‐sided limits [src] 
     / x    \
     |E  - E|
 lim |------|
x->1+\-1 + x/
lim⁡x→1+(ex−ex−1)\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right) 
E
ee 
= 2.71828182845905
     / x    \
     |E  - E|
 lim |------|
x->1-\-1 + x/
lim⁡x→1−(ex−ex−1)\lim_{x \to 1^-}\left(\frac{e^{x} - e}{x - 1}\right) 
E
ee 
= 2.71828182845905
= 2.71828182845905
Rapid solution [src] 
E
ee
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→1−(ex−ex−1)=e\lim_{x \to 1^-}\left(\frac{e^{x} - e}{x - 1}\right) = e
More at x→1 from the left
lim⁡x→1+(ex−ex−1)=e\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right) = e
lim⁡x→∞(ex−ex−1)=∞\lim_{x \to \infty}\left(\frac{e^{x} - e}{x - 1}\right) = \infty
More at x→oo
lim⁡x→0−(ex−ex−1)=−1+e\lim_{x \to 0^-}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e
More at x→0 from the left
lim⁡x→0+(ex−ex−1)=−1+e\lim_{x \to 0^+}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e
More at x→0 from the right
lim⁡x→−∞(ex−ex−1)=0\lim_{x \to -\infty}\left(\frac{e^{x} - e}{x - 1}\right) = 0
More at x→-oo
Numerical answer [src] 
2.71828182845905
2.71828182845905
The graph
Limit of the function (e^x-e)/(-1+x)
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