Limit of the function 4^x/x

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     / x\
     |4 |
 lim |--|
x->oo\x /

\lim_{x \to \infty}\left(\frac{4^{x}}{x}\right)

Limit(4^x/x, x, oo, dir='-')

Lopital's rule

We have indeterminateness of type

oo/oo,

i.e. limit for the numerator is

\lim_{x \to \infty} 4^{x} = \infty

and limit for the denominator is

\lim_{x \to \infty} x = \infty

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{x \to \infty}\left(\frac{4^{x}}{x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 4^{x}}{\frac{d}{d x} x}\right)

\lim_{x \to \infty}\left(2 \cdot 4^{x} \log{\left(2 \right)}\right)

\lim_{x \to \infty}\left(2 \cdot 4^{x} \log{\left(2 \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

Plot the graph

Rapid solution [src]

oo

\infty

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{4^{x}}{x}\right) = \infty

\lim_{x \to 0^-}\left(\frac{4^{x}}{x}\right) = -\infty

\lim_{x \to 0^+}\left(\frac{4^{x}}{x}\right) = \infty

\lim_{x \to 1^-}\left(\frac{4^{x}}{x}\right) = 4

\lim_{x \to 1^+}\left(\frac{4^{x}}{x}\right) = 4

\lim_{x \to -\infty}\left(\frac{4^{x}}{x}\right) = 0

The graph