Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of cot(x)/csc(x)
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Limit of cos(n)/n
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Limit of 21
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Limit of x^2
- Graphing y =:
- cot(x)/csc(x)
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Identical expressions
- cot(x)/csc(x)
- cotangent of (x) divide by csc(x)
- cotx/cscx
- cot(x) divide by csc(x)
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Similar expressions
- (x/sin(x)+cot(x))/csc(x)
- cot(x)/cosec(x)
- cot(x)/cosecx
Limit of the function cot(x)/csc(x)
The solution
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[src]
/cot(x)\ lim |------| x->0+\csc(x)/
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
Limit(cot(x)/csc(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \frac{1}{\csc{\left(x \right)}} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\csc{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{3}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{\csc{\left(x \right)}}{\cot^{3}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(\frac{\left(3 \cot^{2}{\left(x \right)} + 3\right) \csc{\left(x \right)}}{\cot^{4}{\left(x \right)}} - \frac{\csc{\left(x \right)}}{\cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(\frac{\left(3 \cot^{2}{\left(x \right)} + 3\right) \csc{\left(x \right)}}{\cot^{4}{\left(x \right)}} - \frac{\csc{\left(x \right)}}{\cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$1$$
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
$$\lim_{x \to 0^+} \frac{1}{\csc{\left(x \right)}} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\csc{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{3}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot^{2}{\left(x \right)} + 1}}{\frac{d}{d x} \frac{\csc{\left(x \right)}}{\cot^{3}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(\frac{\left(3 \cot^{2}{\left(x \right)} + 3\right) \csc{\left(x \right)}}{\cot^{4}{\left(x \right)}} - \frac{\csc{\left(x \right)}}{\cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)}}{\left(\frac{\left(3 \cot^{2}{\left(x \right)} + 3\right) \csc{\left(x \right)}}{\cot^{4}{\left(x \right)}} - \frac{\csc{\left(x \right)}}{\cot^{2}{\left(x \right)}}\right) \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}\right)$$
=
$$1$$
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]
/cot(x)\ lim |------| x->0+\csc(x)/
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
/cot(x)\ lim |------| x->0-\csc(x)/
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right) = \frac{\sin{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right) = \frac{\sin{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right) = \frac{\sin{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right) = \frac{\sin{\left(1 \right)}}{\tan{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{\csc{\left(x \right)}}\right)$$
More at x→-oo
The graph