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2*x*sin(5*x)/5
Limit of the function/ 2*x*sin(5*x)/5

Limit of the function 2*x*sin(5*x)/5

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     /2*x*sin(5*x)\
 lim |------------|
x->0+\     5      /
$$\lim_{x \to 0^+}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right)$$ 
Limit(((2*x)*sin(5*x))/5, 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
One‐sided limits [src] 
     /2*x*sin(5*x)\
 lim |------------|
x->0+\     5      /
$$\lim_{x \to 0^+}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right)$$ 
0
$$0$$ 
= -5.34536365768728e-30
     /2*x*sin(5*x)\
 lim |------------|
x->0-\     5      /
$$\lim_{x \to 0^-}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right)$$ 
0
$$0$$ 
= -5.34536365768728e-30
= -5.34536365768728e-30
Rapid solution [src] 
0
$$0$$
Expand and simplify
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right) = \frac{2 \sin{\left(5 \right)}}{5}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right) = \frac{2 \sin{\left(5 \right)}}{5}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 x \sin{\left(5 x \right)}}{5}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→-oo
Numerical answer [src] 
-5.34536365768728e-30
-5.34536365768728e-30
The graph
Limit of the function 2*x*sin(5*x)/5
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