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x*sin(5*x)/6
Limit of the function/ x*sin(5*x)/6

Limit of the function x*sin(5*x)/6

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The solution

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     /x*sin(5*x)\
 lim |----------|
x->0+\    6     /
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(5 x \right)}}{6}\right)$$ 
Limit((x*sin(5*x))/6, 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] 
     /x*sin(5*x)\
 lim |----------|
x->0+\    6     /
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(5 x \right)}}{6}\right)$$ 
0
$$0$$ 
= -2.2272348573697e-30
     /x*sin(5*x)\
 lim |----------|
x->0-\    6     /
$$\lim_{x \to 0^-}\left(\frac{x \sin{\left(5 x \right)}}{6}\right)$$ 
0
$$0$$ 
= -2.2272348573697e-30
= -2.2272348573697e-30
Rapid solution [src] 
0
$$0$$
Expand and simplify
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x \sin{\left(5 x \right)}}{6}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(5 x \right)}}{6}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{x \sin{\left(5 x \right)}}{6}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x \sin{\left(5 x \right)}}{6}\right) = \frac{\sin{\left(5 \right)}}{6}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x \sin{\left(5 x \right)}}{6}\right) = \frac{\sin{\left(5 \right)}}{6}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x \sin{\left(5 x \right)}}{6}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→-oo
Numerical answer [src] 
-2.2272348573697e-30
-2.2272348573697e-30
The graph
Limit of the function x*sin(5*x)/6
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