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

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

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     /sin(6*pi*x)\
 lim |-----------|
x->1+\ sin(pi*x) /
lim⁡x→1+(sin⁡(6πx)sin⁡(πx))\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) 
Limit(sin((6*pi)*x)/sin(pi*x), x, 1)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→1+sin⁡(6πx)=0\lim_{x \to 1^+} \sin{\left(6 \pi x \right)} = 0
and limit for the denominator is
lim⁡x→1+sin⁡(πx)=0\lim_{x \to 1^+} \sin{\left(\pi x \right)} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→1+(sin⁡(6πx)sin⁡(πx))\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)
=
Let's transform the function under the limit a few
lim⁡x→1+(sin⁡(6πx)sin⁡(πx))\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)
=
lim⁡x→1+(ddxsin⁡(6πx)ddxsin⁡(πx))\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \sin{\left(6 \pi x \right)}}{\frac{d}{d x} \sin{\left(\pi x \right)}}\right)
=
lim⁡x→1+(6cos⁡(6πx)cos⁡(πx))\lim_{x \to 1^+}\left(\frac{6 \cos{\left(6 \pi x \right)}}{\cos{\left(\pi x \right)}}\right)
=
lim⁡x→1+−6\lim_{x \to 1^+} -6
=
lim⁡x→1+−6\lim_{x \to 1^+} -6
=
−6-6
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
-2.0-1.5-1.0-0.52.00.00.51.01.5-1010
Plot the graph
Rapid solution [src] 
-6
−6-6
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→1−(sin⁡(6πx)sin⁡(πx))=−6\lim_{x \to 1^-}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = -6
More at x→1 from the left
lim⁡x→1+(sin⁡(6πx)sin⁡(πx))=−6\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = -6
lim⁡x→∞(sin⁡(6πx)sin⁡(πx))\lim_{x \to \infty}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)
More at x→oo
lim⁡x→0−(sin⁡(6πx)sin⁡(πx))=6\lim_{x \to 0^-}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = 6
More at x→0 from the left
lim⁡x→0+(sin⁡(6πx)sin⁡(πx))=6\lim_{x \to 0^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = 6
More at x→0 from the right
lim⁡x→−∞(sin⁡(6πx)sin⁡(πx))\lim_{x \to -\infty}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)
More at x→-oo
One‐sided limits [src] 
     /sin(6*pi*x)\
 lim |-----------|
x->1+\ sin(pi*x) /
lim⁡x→1+(sin⁡(6πx)sin⁡(πx))\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) 
-6
−6-6 
= -6.0
     /sin(6*pi*x)\
 lim |-----------|
x->1-\ sin(pi*x) /
lim⁡x→1−(sin⁡(6πx)sin⁡(πx))\lim_{x \to 1^-}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) 
-6
−6-6 
= -6.0
= -6.0
Numerical answer [src] 
-6.0
-6.0
The graph
Limit of the function sin(6*pi*x)/sin(pi*x)
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