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e^(2*x)/x
Limit of the function/ e^(2*x)/x

Limit of the function e^(2*x)/x

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The solution

You have entered [src] 
     / 2*x\
     |E   |
 lim |----|
x->oo\ x  /
lim⁡x→∞(e2xx)\lim_{x \to \infty}\left(\frac{e^{2 x}}{x}\right) 
Limit(E^(2*x)/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞e2x=∞\lim_{x \to \infty} e^{2 x} = \infty
and limit for the denominator is
lim⁡x→∞x=∞\lim_{x \to \infty} x = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(e2xx)\lim_{x \to \infty}\left(\frac{e^{2 x}}{x}\right)
=
Let's transform the function under the limit a few
lim⁡x→∞(e2xx)\lim_{x \to \infty}\left(\frac{e^{2 x}}{x}\right)
=
lim⁡x→∞(ddxe2xddxx)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} e^{2 x}}{\frac{d}{d x} x}\right)
=
lim⁡x→∞(2e2x)\lim_{x \to \infty}\left(2 e^{2 x}\right)
=
lim⁡x→∞(2e2x)\lim_{x \to \infty}\left(2 e^{2 x}\right)
=
∞\infty
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
02468-8-6-4-2-1010-5000000050000000
Plot the graph
Rapid solution [src] 
oo
∞\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(e2xx)=∞\lim_{x \to \infty}\left(\frac{e^{2 x}}{x}\right) = \infty
lim⁡x→0−(e2xx)=−∞\lim_{x \to 0^-}\left(\frac{e^{2 x}}{x}\right) = -\infty
More at x→0 from the left
lim⁡x→0+(e2xx)=∞\lim_{x \to 0^+}\left(\frac{e^{2 x}}{x}\right) = \infty
More at x→0 from the right
lim⁡x→1−(e2xx)=e2\lim_{x \to 1^-}\left(\frac{e^{2 x}}{x}\right) = e^{2}
More at x→1 from the left
lim⁡x→1+(e2xx)=e2\lim_{x \to 1^+}\left(\frac{e^{2 x}}{x}\right) = e^{2}
More at x→1 from the right
lim⁡x→−∞(e2xx)=0\lim_{x \to -\infty}\left(\frac{e^{2 x}}{x}\right) = 0
More at x→-oo
The graph
Limit of the function e^(2*x)/x
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