Mister Exam

Other calculators:

(-1+e^(3*x))/x
Limit of the function/ (-1+e^(3*x))/x

Limit of the function (-1+e^(3*x))/x

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /      3*x\
     |-1 + E   |
 lim |---------|
x->0+\    x    /
lim⁡x→0+(e3x−1x)\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right) 
Limit((-1 + E^(3*x))/x, x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+(e3x−1)=0\lim_{x \to 0^+}\left(e^{3 x} - 1\right) = 0
and limit for the denominator is
lim⁡x→0+x=0\lim_{x \to 0^+} x = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(e3x−1x)\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(e3x−1x)\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right)
=
lim⁡x→0+(ddx(e3x−1)ddxx)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(e^{3 x} - 1\right)}{\frac{d}{d x} x}\right)
=
lim⁡x→0+(3e3x)\lim_{x \to 0^+}\left(3 e^{3 x}\right)
=
lim⁡x→0+3\lim_{x \to 0^+} 3
=
lim⁡x→0+3\lim_{x \to 0^+} 3
=
33
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
02468-8-6-4-2-101002000000000000
Plot the graph
One‐sided limits [src] 
     /      3*x\
     |-1 + E   |
 lim |---------|
x->0+\    x    /
lim⁡x→0+(e3x−1x)\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right) 
3
33 
= 3.0
     /      3*x\
     |-1 + E   |
 lim |---------|
x->0-\    x    /
lim⁡x→0−(e3x−1x)\lim_{x \to 0^-}\left(\frac{e^{3 x} - 1}{x}\right) 
3
33 
= 3.0
= 3.0
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(e3x−1x)=3\lim_{x \to 0^-}\left(\frac{e^{3 x} - 1}{x}\right) = 3
More at x→0 from the left
lim⁡x→0+(e3x−1x)=3\lim_{x \to 0^+}\left(\frac{e^{3 x} - 1}{x}\right) = 3
lim⁡x→∞(e3x−1x)=∞\lim_{x \to \infty}\left(\frac{e^{3 x} - 1}{x}\right) = \infty
More at x→oo
lim⁡x→1−(e3x−1x)=−1+e3\lim_{x \to 1^-}\left(\frac{e^{3 x} - 1}{x}\right) = -1 + e^{3}
More at x→1 from the left
lim⁡x→1+(e3x−1x)=−1+e3\lim_{x \to 1^+}\left(\frac{e^{3 x} - 1}{x}\right) = -1 + e^{3}
More at x→1 from the right
lim⁡x→−∞(e3x−1x)=0\lim_{x \to -\infty}\left(\frac{e^{3 x} - 1}{x}\right) = 0
More at x→-oo
Rapid solution [src] 
3
33
Expand and simplify
Numerical answer [src] 
3.0
3.0
The graph
Limit of the function (-1+e^(3*x))/x
    © Mister Exam – Calculator