Mister Exam
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How to use it?
- Limit of the function:
-
Limit of (-6+x^2-x)/(6+x^2-5*x)
-
Limit of log(1+x^2)/(-e^(-x)+cos(3*x))
-
Limit of -2+|-2+x|/x
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Limit of (-4-8*x+8*x^2)/(3+8*x)
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Identical expressions
- (x- two *acot(x))/x
- (x minus 2 multiply by arcco tangent of gent of (x)) divide by x
- (x minus two multiply by arcco tangent of gent of (x)) divide by x
- (x-2acot(x))/x
- x-2acotx/x
- (x-2*acot(x)) divide by x
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Similar expressions
- (x+2*acot(x))/x
- (x-2*arccot(x))/x
- (x-2*arccotx)/x
Limit of the function (x-2*acot(x))/x
The solution
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/x - 2*acot(x)\ lim |-------------| x->oo\ x /
$$\lim_{x \to \infty}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right)$$
Limit((x - 2*acot(x))/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x - 2 \operatorname{acot}{\left(x \right)}\right) = \infty$$
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.
$$\lim_{x \to \infty}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x - 2 \operatorname{acot}{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(1 + \frac{2}{x^{2} + 1}\right)$$
=
$$\lim_{x \to \infty}\left(1 + \frac{2}{x^{2} + 1}\right)$$
=
$$1$$
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)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x - 2 \operatorname{acot}{\left(x \right)}\right) = \infty$$
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.
$$\lim_{x \to \infty}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x - 2 \operatorname{acot}{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(1 + \frac{2}{x^{2} + 1}\right)$$
=
$$\lim_{x \to \infty}\left(1 + \frac{2}{x^{2} + 1}\right)$$
=
$$1$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to 0^-}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = 1 - \frac{\pi}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = 1 - \frac{\pi}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = 1$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = 1 - \frac{\pi}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = 1 - \frac{\pi}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 2 \operatorname{acot}{\left(x \right)}}{x}\right) = 1$$
More at x→-oo
The graph