Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of sin(x)/x+tan(x)
-
Limit of -6+x^2+9*x/2
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Limit of x^2*sin(x^(-3))
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Limit of 16-16*x-13*x^2/3
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Identical expressions
- x^(- two)-cot(x)/x
- x to the power of ( minus 2) minus cotangent of (x) divide by x
- x to the power of ( minus two) minus cotangent of (x) divide by x
- x(-2)-cot(x)/x
- x-2-cotx/x
- x^-2-cotx/x
- x^(-2)-cot(x) divide by x
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Similar expressions
- x^(-2)+cot(x)/x
- x^(2)-cot(x)/x
Limit of the function x^(-2)-cot(x)/x
The solution
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[src]
/1 cot(x)\
lim |-- - ------|
x->0+| 2 x |
\x /
$$\lim_{x \to 0^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
Limit(x^(-2) - cot(x)/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- x \cot{\left(x \right)} + 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{- x \cot{\left(x \right)} + 1}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \cot{\left(x \right)} + 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x \cot^{2}{\left(x \right)} + x - \cot{\left(x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x \cot^{2}{\left(x \right)} + x - \cot{\left(x \right)}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- x \cot^{3}{\left(x \right)} - x \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)$$
=
$$\lim_{x \to 0^+}\left(- x \cot^{3}{\left(x \right)} - x \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)$$
=
$$\frac{1}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(- x \cot{\left(x \right)} + 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{- x \cot{\left(x \right)} + 1}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- x \cot{\left(x \right)} + 1\right)}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{x \cot^{2}{\left(x \right)} + x - \cot{\left(x \right)}}{2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(x \cot^{2}{\left(x \right)} + x - \cot{\left(x \right)}\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+}\left(- x \cot^{3}{\left(x \right)} - x \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)$$
=
$$\lim_{x \to 0^+}\left(- x \cot^{3}{\left(x \right)} - x \cot{\left(x \right)} + \cot^{2}{\left(x \right)} + 1\right)$$
=
$$\frac{1}{3}$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right) = \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right) = \frac{-1 + \tan{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right) = \frac{-1 + \tan{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right) = \frac{1}{3}$$
$$\lim_{x \to \infty}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right) = \frac{-1 + \tan{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right) = \frac{-1 + \tan{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
More at x→-oo
One‐sided limits
[src]
/1 cot(x)\
lim |-- - ------|
x->0+| 2 x |
\x /
$$\lim_{x \to 0^+}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
/1 cot(x)\
lim |-- - ------|
x->0-| 2 x |
\x /
$$\lim_{x \to 0^-}\left(- \frac{\cot{\left(x \right)}}{x} + \frac{1}{x^{2}}\right)$$
1/3
$$\frac{1}{3}$$
= 0.333333333333333
= 0.333333333333333
The graph