Mister Exam

Other calculators:

tan(8*x)/sin(5*x)
Limit of the function/ tan(8*x)/sin(5*x)

Limit of the function tan(8*x)/sin(5*x)

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /tan(8*x)\
 lim |--------|
x->0+\sin(5*x)/
lim⁡x→0+(tan⁡(8x)sin⁡(5x))\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) 
Limit(tan(8*x)/sin(5*x), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+tan⁡(8x)=0\lim_{x \to 0^+} \tan{\left(8 x \right)} = 0
and limit for the denominator is
lim⁡x→0+sin⁡(5x)=0\lim_{x \to 0^+} \sin{\left(5 x \right)} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(tan⁡(8x)sin⁡(5x))\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)
=
lim⁡x→0+(ddxtan⁡(8x)ddxsin⁡(5x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(8 x \right)}}{\frac{d}{d x} \sin{\left(5 x \right)}}\right)
=
lim⁡x→0+(8tan⁡2(8x)+85cos⁡(5x))\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)} + 8}{5 \cos{\left(5 x \right)}}\right)
=
lim⁡x→0+(8tan⁡2(8x)5+85)\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)}}{5} + \frac{8}{5}\right)
=
lim⁡x→0+(8tan⁡2(8x)5+85)\lim_{x \to 0^+}\left(\frac{8 \tan^{2}{\left(8 x \right)}}{5} + \frac{8}{5}\right)
=
85\frac{8}{5}
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
02468-8-6-4-2-1010-200200
Plot the graph
One‐sided limits [src] 
     /tan(8*x)\
 lim |--------|
x->0+\sin(5*x)/
lim⁡x→0+(tan⁡(8x)sin⁡(5x))\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) 
8/5
85\frac{8}{5} 
= 1.6
     /tan(8*x)\
 lim |--------|
x->0-\sin(5*x)/
lim⁡x→0−(tan⁡(8x)sin⁡(5x))\lim_{x \to 0^-}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) 
8/5
85\frac{8}{5} 
= 1.6
= 1.6
Rapid solution [src] 
8/5
85\frac{8}{5}
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(tan⁡(8x)sin⁡(5x))=85\lim_{x \to 0^-}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{8}{5}
More at x→0 from the left
lim⁡x→0+(tan⁡(8x)sin⁡(5x))=85\lim_{x \to 0^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{8}{5}
lim⁡x→∞(tan⁡(8x)sin⁡(5x))\lim_{x \to \infty}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)
More at x→oo
lim⁡x→1−(tan⁡(8x)sin⁡(5x))=tan⁡(8)sin⁡(5)\lim_{x \to 1^-}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\sin{\left(5 \right)}}
More at x→1 from the left
lim⁡x→1+(tan⁡(8x)sin⁡(5x))=tan⁡(8)sin⁡(5)\lim_{x \to 1^+}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right) = \frac{\tan{\left(8 \right)}}{\sin{\left(5 \right)}}
More at x→1 from the right
lim⁡x→−∞(tan⁡(8x)sin⁡(5x))\lim_{x \to -\infty}\left(\frac{\tan{\left(8 x \right)}}{\sin{\left(5 x \right)}}\right)
More at x→-oo
Numerical answer [src] 
1.6
1.6
The graph
Limit of the function tan(8*x)/sin(5*x)
    © Mister Exam – Calculator