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Limit of the function (x-atan(x))/x

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     /x - atan(x)\
 lim |-----------|
x->oo\     x     /

\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x}\right)

Limit((x - atan(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 - \operatorname{atan}{\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(\frac{x - \operatorname{atan}{\left(x \right)}}{x}\right)

\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x - \operatorname{atan}{\left(x \right)}\right)}{\frac{d}{d x} x}\right)

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

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

1

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

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Rapid solution [src]

1

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to \infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x}\right) = 1

\lim_{x \to 0^-}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x}\right) = 0

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

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

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

\lim_{x \to -\infty}\left(\frac{x - \operatorname{atan}{\left(x \right)}}{x}\right) = 1

The graph