Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (e^x-e)/(-1+x)
-
Limit of (1-3*x)^(1/x)
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Limit of pi/(x*cot(pi*x/2))
-
Limit of sin(7*x)/x^2
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Identical expressions
- pi/(x*cot(pi*x/ two))
- Pi divide by (x multiply by cotangent of ( Pi multiply by x divide by 2))
- Pi divide by (x multiply by cotangent of ( Pi multiply by x divide by two))
- pi/(xcot(pix/2))
- pi/xcotpix/2
- pi divide by (x*cot(pi*x divide by 2))
Limit of the function pi/(x*cot(pi*x/2))
The solution
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[src]
/ pi \
lim |-----------|
x->0+| /pi*x\|
|x*cot|----||
\ \ 2 //
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
Limit(pi/((x*cot((pi*x)/2))), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(\frac{\pi x}{2} \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(\frac{x}{\pi}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(\frac{\pi x}{2} \right)}}}{\frac{d}{d x} \frac{x}{\pi}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\pi^{2} \left(- \cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)}{2 \cot^{2}{\left(\frac{\pi x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\pi^{2} \left(- \cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)}{2 \cot^{2}{\left(\frac{\pi x}{2} \right)}}\right)$$
=
$$\frac{\pi^{2}}{2}$$
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
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(\frac{\pi x}{2} \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(\frac{x}{\pi}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\cot{\left(\frac{\pi x}{2} \right)}}}{\frac{d}{d x} \frac{x}{\pi}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\pi^{2} \left(- \cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)}{2 \cot^{2}{\left(\frac{\pi x}{2} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\pi^{2} \left(- \cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)}{2 \cot^{2}{\left(\frac{\pi x}{2} \right)}}\right)$$
=
$$\frac{\pi^{2}}{2}$$
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]
/ pi \
lim |-----------|
x->0+| /pi*x\|
|x*cot|----||
\ \ 2 //
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
2 pi --- 2
$$\frac{\pi^{2}}{2}$$
= 4.93480220054468
/ pi \
lim |-----------|
x->0-| /pi*x\|
|x*cot|----||
\ \ 2 //
$$\lim_{x \to 0^-}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
2 pi --- 2
$$\frac{\pi^{2}}{2}$$
= 4.93480220054468
= 4.93480220054468
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right) = \frac{\pi^{2}}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right) = \frac{\pi^{2}}{2}$$
$$\lim_{x \to \infty}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right) = \frac{\pi^{2}}{2}$$
$$\lim_{x \to \infty}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\pi}{x \cot{\left(\frac{\pi x}{2} \right)}}\right)$$
More at x→-oo
The graph