Mister Exam
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How to use it?
- Limit of the function:
-
Limit of cos(x^2)
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Limit of (-sin(x)+cos(x))^(1/x)
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Limit of (cos(x)/cosh(x))^(1/(x*log(1+x)))
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Limit of (-1+cos(x))*cot(x)
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Identical expressions
- (- one +cos(x))*cot(x)
- ( minus 1 plus co sinus of e of (x)) multiply by cotangent of (x)
- ( minus one plus co sinus of e of (x)) multiply by cotangent of (x)
- (-1+cos(x))cot(x)
- -1+cosxcotx
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Similar expressions
- (-1-cos(x))*cot(x)
- (1+cos(x))*cot(x)
- (-1+cosx)*cot(x)
Limit of the function (-1+cos(x))*cot(x)
The solution
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lim ((-1 + cos(x))*cot(x)) x->0+
$$\lim_{x \to 0^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
Limit((-1 + cos(x))*cot(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\cos{\left(x \right)} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\cos{\left(x \right)} - 1\right)}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \left(- \frac{1}{\sin{\left(x \right)} \cot^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \frac{2 \cot^{2}{\left(x \right)} + 2}{\sin{\left(x \right)} \cot^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \frac{2 \cot^{2}{\left(x \right)} + 2}{\sin{\left(x \right)} \cot^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+}\left(\cos{\left(x \right)} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\cos{\left(x \right)} - 1\right)}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)} \cot^{2}{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \left(- \frac{1}{\sin{\left(x \right)} \cot^{2}{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \frac{2 \cot^{2}{\left(x \right)} + 2}{\sin{\left(x \right)} \cot^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(- \frac{2 \cot^{2}{\left(x \right)} + 2}{\sin{\left(x \right)} \cot^{3}{\left(x \right)}} + \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$0$$
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
One‐sided limits
[src]
lim ((-1 + cos(x))*cot(x)) x->0+
$$\lim_{x \to 0^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
0
$$0$$
= 9.41021530176551e-32
lim ((-1 + cos(x))*cot(x)) x->0-
$$\lim_{x \to 0^-}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
0
$$0$$
= -9.41021530176551e-32
= -9.41021530176551e-32
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right) = \frac{-1 + \cos{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right) = \frac{-1 + \cos{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right) = \frac{-1 + \cos{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right) = \frac{-1 + \cos{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\left(\cos{\left(x \right)} - 1\right) \cot{\left(x \right)}\right)$$
More at x→-oo
The graph