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sin(7*x)/tan(3*x)^2

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  • Limit of the function:
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  • Identical expressions

  • sin(seven *x)/tan(three *x)^ two
  • sinus of (7 multiply by x) divide by tangent of (3 multiply by x) squared
  • sinus of (seven multiply by x) divide by tangent of (three multiply by x) to the power of two
  • sin(7*x)/tan(3*x)2
  • sin7*x/tan3*x2
  • sin(7*x)/tan(3*x)²
  • sin(7*x)/tan(3*x) to the power of 2
  • sin(7x)/tan(3x)^2
  • sin(7x)/tan(3x)2
  • sin7x/tan3x2
  • sin7x/tan3x^2
  • sin(7*x) divide by tan(3*x)^2
Limit of the function/ sin(7*x)/tan(3*x)^2

Limit of the function sin(7*x)/tan(3*x)^2

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The solution

You have entered [src] 
     / sin(7*x)\
 lim |---------|
x->0+|   2     |
     \tan (3*x)/
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right)$$ 
Limit(sin(7*x)/(tan(3*x)^2), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(7 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \tan^{2}{\left(3 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(7 x \right)}}{\frac{d}{d x} \tan^{2}{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cos{\left(7 x \right)}}{\left(6 \tan^{2}{\left(3 x \right)} + 6\right) \tan{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7}{6 \tan{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7}{6 \tan{\left(3 x \right)}}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right) = \frac{\sin{\left(7 \right)}}{\tan^{2}{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right) = \frac{\sin{\left(7 \right)}}{\tan^{2}{\left(3 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
More at x→-oo
One‐sided limits [src] 
     / sin(7*x)\
 lim |---------|
x->0+|   2     |
     \tan (3*x)/
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right)$$ 
oo
$$\infty$$ 
= 117.371490930519
     / sin(7*x)\
 lim |---------|
x->0-|   2     |
     \tan (3*x)/
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(7 x \right)}}{\tan^{2}{\left(3 x \right)}}\right)$$ 
-oo
$$-\infty$$ 
= -117.371490930519
= -117.371490930519
Numerical answer [src] 
117.371490930519
117.371490930519
The graph
Limit of the function sin(7*x)/tan(3*x)^2
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