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x*sin(2/x)
Limit of the function/ x*sin(2/x)

Limit of the function x*sin(2/x)

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     /     /2\\
 lim |x*sin|-||
x->0+\     \x//
$$\lim_{x \to 0^+}\left(x \sin{\left(\frac{2}{x} \right)}\right)$$ 
Limit(x*sin(2/x), 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
Plot the graph
Rapid solution [src] 
0
$$0$$
Expand and simplify
One‐sided limits [src] 
     /     /2\\
 lim |x*sin|-||
x->0+\     \x//
$$\lim_{x \to 0^+}\left(x \sin{\left(\frac{2}{x} \right)}\right)$$ 
0
$$0$$ 
= 9.68296175008847e-21
     /     /2\\
 lim |x*sin|-||
x->0-\     \x//
$$\lim_{x \to 0^-}\left(x \sin{\left(\frac{2}{x} \right)}\right)$$ 
0
$$0$$ 
= 9.68296175008847e-21
= 9.68296175008847e-21
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(x \sin{\left(\frac{2}{x} \right)}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x \sin{\left(\frac{2}{x} \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(x \sin{\left(\frac{2}{x} \right)}\right) = 2$$
More at x→oo
$$\lim_{x \to 1^-}\left(x \sin{\left(\frac{2}{x} \right)}\right) = \sin{\left(2 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x \sin{\left(\frac{2}{x} \right)}\right) = \sin{\left(2 \right)}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x \sin{\left(\frac{2}{x} \right)}\right) = 2$$
More at x→-oo
Numerical answer [src] 
9.68296175008847e-21
9.68296175008847e-21
The graph
Limit of the function x*sin(2/x)
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