Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
-
Limit of (1+2*n)/(-1+3*n)
-
Limit of (x^3-3^x)/(-3+x)
-
Limit of -x+(-2+x)^4/(3+x)^4
- Graphing y =:
- x*cot(x/2)
-
Identical expressions
- x*cot(x/ two)
- x multiply by cotangent of (x divide by 2)
- x multiply by cotangent of (x divide by two)
- xcot(x/2)
- xcotx/2
- x*cot(x divide by 2)
Limit of the function x*cot(x/2)
The solution
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[src]
/ /x\\ lim |x*cot|-|| x->0+\ \2//
$$\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
Limit(x*cot(x/2), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(\frac{x}{2} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\cot{\left(\frac{x}{2} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}\right)$$
=
$$2$$
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
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \frac{1}{\cot{\left(\frac{x}{2} \right)}} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \frac{1}{\cot{\left(\frac{x}{2} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}}\right)$$
=
$$2$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(x \cot{\left(\frac{x}{2} \right)}\right) = 2$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(x \cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\tan{\left(\frac{1}{2} \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\tan{\left(\frac{1}{2} \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) = 2$$
$$\lim_{x \to \infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(x \cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\tan{\left(\frac{1}{2} \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x \cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\tan{\left(\frac{1}{2} \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
More at x→-oo
One‐sided limits
[src]
/ /x\\ lim |x*cot|-|| x->0+\ \2//
$$\lim_{x \to 0^+}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
2
$$2$$
= 2.0
/ /x\\ lim |x*cot|-|| x->0-\ \2//
$$\lim_{x \to 0^-}\left(x \cot{\left(\frac{x}{2} \right)}\right)$$
2
$$2$$
= 2.0
= 2.0
The graph