Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (4-x^2)/(3-x^2)
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Limit of (-2+x^2-x)/(-2+x+3*x^2)
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Limit of 5-9*x+3*x^2/2
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Limit of (-16+x^2)/(-64+x^3)
- Integral of d{x}:
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x*sin(x/2)
- Derivative of:
- x*sin(x/2)
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Identical expressions
- x*sin(x/ two)
- x multiply by sinus of (x divide by 2)
- x multiply by sinus of (x divide by two)
- xsin(x/2)
- xsinx/2
- x*sin(x divide by 2)
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Similar expressions
- 4*cos(8*x)*sin(x/2)/x
- (1-cos(2*x)+x*sin(x))/(2*x*sin(x))
- cos(x)*sin(x)/(2*x)
Limit of the function x*sin(x/2)
The solution
You have entered
[src]
/ /x\\ lim |x*sin|-|| x->0+\ \2//
$$\lim_{x \to 0^+}\left(x \sin{\left(\frac{x}{2} \right)}\right)$$
Limit(x*sin(x/2), x, 0)
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(x \sin{\left(\frac{x}{2} \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x \sin{\left(\frac{x}{2} \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(x \sin{\left(\frac{x}{2} \right)}\right) = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→oo
$$\lim_{x \to 1^-}\left(x \sin{\left(\frac{x}{2} \right)}\right) = \sin{\left(\frac{1}{2} \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x \sin{\left(\frac{x}{2} \right)}\right) = \sin{\left(\frac{1}{2} \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x \sin{\left(\frac{x}{2} \right)}\right) = - \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x \sin{\left(\frac{x}{2} \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(x \sin{\left(\frac{x}{2} \right)}\right) = \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→oo
$$\lim_{x \to 1^-}\left(x \sin{\left(\frac{x}{2} \right)}\right) = \sin{\left(\frac{1}{2} \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x \sin{\left(\frac{x}{2} \right)}\right) = \sin{\left(\frac{1}{2} \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x \sin{\left(\frac{x}{2} \right)}\right) = - \infty \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$
More at x→-oo
One‐sided limits
[src]
/ /x\\ lim |x*sin|-|| x->0+\ \2//
$$\lim_{x \to 0^+}\left(x \sin{\left(\frac{x}{2} \right)}\right)$$
0
$$0$$
= -1.326657548588e-32
/ /x\\ lim |x*sin|-|| x->0-\ \2//
$$\lim_{x \to 0^-}\left(x \sin{\left(\frac{x}{2} \right)}\right)$$
0
$$0$$
= -1.326657548588e-32
= -1.326657548588e-32
The graph