Mister Exam
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- Limit of the function:
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Limit of (1+(1+n)^2)/(1+n^2)
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Limit of cot(2*x)/cot(8*x)
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Limit of tan(1/x)/x
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Limit of cos(pi*x)
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Identical expressions
- cot(two *x)/cot(eight *x)
- cotangent of (2 multiply by x) divide by cotangent of (8 multiply by x)
- cotangent of (two multiply by x) divide by cotangent of (eight multiply by x)
- cot(2x)/cot(8x)
- cot2x/cot8x
- cot(2*x) divide by cot(8*x)
Limit of the function cot(2*x)/cot(8*x)
The solution
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/cot(2*x)\ lim |--------| x->0+\cot(8*x)/
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
Limit(cot(2*x)/cot(8*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(8 x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(2 x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(8 x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}}{\left(2 \cot^{2}{\left(2 x \right)} + 2\right) \cot^{2}{\left(8 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(2 x \right)} + 2}}{\frac{d}{d x} \frac{\cot^{2}{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)}}{\left(\frac{\left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}} + \frac{\left(- 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} \cot^{2}{\left(8 x \right)} - 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} - 8 \left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot^{2}{\left(2 x \right)} \cot{\left(8 x \right)}\right) \cot^{2}{\left(8 x \right)}}{\left(8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}\right)^{2}}\right) \left(2 \cot^{2}{\left(2 x \right)} + 2\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)}}{\left(\frac{\left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}} + \frac{\left(- 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} \cot^{2}{\left(8 x \right)} - 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} - 8 \left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot^{2}{\left(2 x \right)} \cot{\left(8 x \right)}\right) \cot^{2}{\left(8 x \right)}}{\left(8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}\right)^{2}}\right) \left(2 \cot^{2}{\left(2 x \right)} + 2\right)^{2}}\right)$$
=
$$4$$
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
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(8 x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(2 x \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(8 x \right)}}}{\frac{d}{d x} \frac{1}{\cot{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}}{\left(2 \cot^{2}{\left(2 x \right)} + 2\right) \cot^{2}{\left(8 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{2 \cot^{2}{\left(2 x \right)} + 2}}{\frac{d}{d x} \frac{\cot^{2}{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)}}{\left(\frac{\left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}} + \frac{\left(- 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} \cot^{2}{\left(8 x \right)} - 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} - 8 \left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot^{2}{\left(2 x \right)} \cot{\left(8 x \right)}\right) \cot^{2}{\left(8 x \right)}}{\left(8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}\right)^{2}}\right) \left(2 \cot^{2}{\left(2 x \right)} + 2\right)^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{2 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)}}{\left(\frac{\left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot{\left(8 x \right)}}{8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}} + \frac{\left(- 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} \cot^{2}{\left(8 x \right)} - 8 \left(- 4 \cot^{2}{\left(2 x \right)} - 4\right) \cot{\left(2 x \right)} - 8 \left(- 16 \cot^{2}{\left(8 x \right)} - 16\right) \cot^{2}{\left(2 x \right)} \cot{\left(8 x \right)}\right) \cot^{2}{\left(8 x \right)}}{\left(8 \cot^{2}{\left(2 x \right)} \cot^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(2 x \right)}\right)^{2}}\right) \left(2 \cot^{2}{\left(2 x \right)} + 2\right)^{2}}\right)$$
=
$$4$$
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(2*x)\ lim |--------| x->0+\cot(8*x)/
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
4
$$4$$
= 4.0
/cot(2*x)\ lim |--------| x->0-\cot(8*x)/
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
4
$$4$$
= 4.0
= 4.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = 4$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\tan{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\tan{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\tan{\left(2 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\tan{\left(2 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(2 x \right)}}{\cot{\left(8 x \right)}}\right)$$
More at x→-oo
The graph