Mister Exam

Other calculators:


(e^x-e)/(-1+x)

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

at
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src]
     / x    \
     |E  - E|
 lim |------|
x->1+\-1 + x/
limx1+(exex1)\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
limx1+(exe)=0\lim_{x \to 1^+}\left(e^{x} - e\right) = 0
and limit for the denominator is
limx1+(x1)=0\lim_{x \to 1^+}\left(x - 1\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
limx1+(exex1)\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)
=
Let's transform the function under the limit a few
limx1+(exex1)\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)
=
limx1+(ddx(exe)ddx(x1))\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(e^{x} - e\right)}{\frac{d}{d x} \left(x - 1\right)}\right)
=
limx1+ex\lim_{x \to 1^+} e^{x}
=
limx1+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
One‐sided limits [src]
     / x    \
     |E  - E|
 lim |------|
x->1+\-1 + x/
limx1+(exex1)\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right)
E
ee
= 2.71828182845905
     / x    \
     |E  - E|
 lim |------|
x->1-\-1 + x/
limx1(exex1)\lim_{x \to 1^-}\left(\frac{e^{x} - e}{x - 1}\right)
E
ee
= 2.71828182845905
= 2.71828182845905
Rapid solution [src]
E
ee
Other limits x→0, -oo, +oo, 1
limx1(exex1)=e\lim_{x \to 1^-}\left(\frac{e^{x} - e}{x - 1}\right) = e
More at x→1 from the left
limx1+(exex1)=e\lim_{x \to 1^+}\left(\frac{e^{x} - e}{x - 1}\right) = e
limx(exex1)=\lim_{x \to \infty}\left(\frac{e^{x} - e}{x - 1}\right) = \infty
More at x→oo
limx0(exex1)=1+e\lim_{x \to 0^-}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e
More at x→0 from the left
limx0+(exex1)=1+e\lim_{x \to 0^+}\left(\frac{e^{x} - e}{x - 1}\right) = -1 + e
More at x→0 from the right
limx(exex1)=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)