Mister Exam

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Limit of the function x+log(x)/x

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     /    log(x)\
 lim |x + ------|
x->oo\      x   /

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

Limit(x + log(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(x^{2} + \log{\left(x \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(x + \frac{\log{\left(x \right)}}{x}\right)

=
Let's transform the function under the limit a few

\lim_{x \to \infty}\left(\frac{x^{2} + \log{\left(x \right)}}{x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{2} + \log{\left(x \right)}\right)}{\frac{d}{d x} x}\right)

\lim_{x \to \infty}\left(2 x + \frac{1}{x}\right)

\lim_{x \to \infty}\left(2 x + \frac{1}{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

Plot the graph

Other limits x→0, -oo, +oo, 1

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

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

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

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

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

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

Rapid solution [src]

oo

\infty

Expand and simplify

The graph