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sin(2*x)/sin(x)^2

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  • Similar expressions

  • sin(2*x)/sinx^2
Limit of the function/ sin(2*x)/sin(x)^2

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

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The solution

You have entered [src] 
     /sin(2*x)\
 lim |--------|
x->0+|   2    |
     \sin (x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$ 
Limit(sin(2*x)/sin(x)^2, x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin^{2}{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(2 x \right)}}{\frac{d}{d x} \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(2 x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{\sin{\left(x \right)}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\sin{\left(x \right)}}$$
=
$$\infty$$
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
Plot the graph
Rapid solution [src] 
oo
$$\infty$$
Expand and simplify
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right) = \frac{\sin{\left(2 \right)}}{\sin^{2}{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right) = \frac{\sin{\left(2 \right)}}{\sin^{2}{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$
More at x→-oo
One‐sided limits [src] 
     /sin(2*x)\
 lim |--------|
x->0+|   2    |
     \sin (x) /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$ 
oo
$$\infty$$ 
= 301.995584976054
     /sin(2*x)\
 lim |--------|
x->0-|   2    |
     \sin (x) /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}}\right)$$ 
-oo
$$-\infty$$ 
= -301.995584976054
= -301.995584976054
Numerical answer [src] 
301.995584976054
301.995584976054
The graph
Limit of the function sin(2*x)/sin(x)^2
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