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Limit of the function log(3*x)/x

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     /log(3*x)\
 lim |--------|
x->oo\   x    /

\lim_{x \to \infty}\left(\frac{\log{\left(3 x \right)}}{x}\right)

Limit(log(3*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}\left(\log{\left(x \right)} + \log{\left(3 \right)}\right) = \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{\log{\left(3 x \right)}}{x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\log{\left(x \right)} + \log{\left(3 \right)}\right)}{\frac{d}{d x} x}\right)

\lim_{x \to \infty} \frac{1}{x}

\lim_{x \to \infty} \frac{1}{x}

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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

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{\log{\left(3 x \right)}}{x}\right) = 0

\lim_{x \to 0^-}\left(\frac{\log{\left(3 x \right)}}{x}\right) = \infty

\lim_{x \to 0^+}\left(\frac{\log{\left(3 x \right)}}{x}\right) = -\infty

\lim_{x \to 1^-}\left(\frac{\log{\left(3 x \right)}}{x}\right) = \log{\left(3 \right)}

\lim_{x \to 1^+}\left(\frac{\log{\left(3 x \right)}}{x}\right) = \log{\left(3 \right)}

\lim_{x \to -\infty}\left(\frac{\log{\left(3 x \right)}}{x}\right) = 0

Rapid solution [src]

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Expand and simplify

The graph