Mister Exam

Other calculators:

5^x/x
Limit of the function/ 5^x/x

Limit of the function 5^x/x

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     / x\
     |5 |
 lim |--|
x->oo\x /
lim⁡x→∞(5xx)\lim_{x \to \infty}\left(\frac{5^{x}}{x}\right) 
Limit(5^x/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞5x=∞\lim_{x \to \infty} 5^{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→∞(5xx)\lim_{x \to \infty}\left(\frac{5^{x}}{x}\right)
=
lim⁡x→∞(ddx5xddxx)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 5^{x}}{\frac{d}{d x} x}\right)
=
lim⁡x→∞(5xlog⁡(5))\lim_{x \to \infty}\left(5^{x} \log{\left(5 \right)}\right)
=
lim⁡x→∞(5xlog⁡(5))\lim_{x \to \infty}\left(5^{x} \log{\left(5 \right)}\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
02468-8-6-4-2-1010-10000001000000
Plot the graph
Rapid solution [src] 
oo
∞\infty
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(5xx)=∞\lim_{x \to \infty}\left(\frac{5^{x}}{x}\right) = \infty
lim⁡x→0−(5xx)=−∞\lim_{x \to 0^-}\left(\frac{5^{x}}{x}\right) = -\infty
More at x→0 from the left
lim⁡x→0+(5xx)=∞\lim_{x \to 0^+}\left(\frac{5^{x}}{x}\right) = \infty
More at x→0 from the right
lim⁡x→1−(5xx)=5\lim_{x \to 1^-}\left(\frac{5^{x}}{x}\right) = 5
More at x→1 from the left
lim⁡x→1+(5xx)=5\lim_{x \to 1^+}\left(\frac{5^{x}}{x}\right) = 5
More at x→1 from the right
lim⁡x→−∞(5xx)=0\lim_{x \to -\infty}\left(\frac{5^{x}}{x}\right) = 0
More at x→-oo
The graph
Limit of the function 5^x/x
    © Mister Exam – Calculator