Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of tan(15*x)/sin(3*x)
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Limit of cot(x)/(pi-2*x)
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Limit of exp(n)
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Limit of sin(3*x)/sin(x)
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Identical expressions
- cot(x)/(pi- two *x)
- cotangent of (x) divide by ( Pi minus 2 multiply by x)
- cotangent of (x) divide by ( Pi minus two multiply by x)
- cot(x)/(pi-2x)
- cotx/pi-2x
- cot(x) divide by (pi-2*x)
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Similar expressions
- cot(x)/(pi+2*x)
Limit of the function cot(x)/(pi-2*x)
The solution
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/ cot(x) \ lim |--------| pi \pi - 2*x/ x->--+ 2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
Limit(cot(x)/(pi - 2*x), x, pi/2)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \frac{\pi}{2}^+} \cot{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to \frac{\pi}{2}^+}\left(- 2 x + \pi\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \cot{\left(x \right)}}{\frac{d}{d x} \left(- 2 x + \pi\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)$$
=
$$\frac{1}{2}$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to \frac{\pi}{2}^+} \cot{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to \frac{\pi}{2}^+}\left(- 2 x + \pi\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\frac{d}{d x} \cot{\left(x \right)}}{\frac{d}{d x} \left(- 2 x + \pi\right)}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)$$
=
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot^{2}{\left(x \right)}}{2} + \frac{1}{2}\right)$$
=
$$\frac{1}{2}$$
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
One‐sided limits
[src]
/ cot(x) \ lim |--------| pi \pi - 2*x/ x->--+ 2
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
/ cot(x) \ lim |--------| pi \pi - 2*x/ x->--- 2
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
= 0.5
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \frac{\pi}{2}^-}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \frac{1}{2}$$
More at x→pi/2 from the left
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \frac{\cot{\left(1 \right)}}{-2 + \pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \frac{\cot{\left(1 \right)}}{-2 + \pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
More at x→-oo
More at x→pi/2 from the left
$$\lim_{x \to \frac{\pi}{2}^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \frac{\cot{\left(1 \right)}}{-2 + \pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right) = \frac{\cot{\left(1 \right)}}{-2 + \pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{- 2 x + \pi}\right)$$
More at x→-oo
The graph