Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of sin(6*pi*x)/sin(pi*x)
- Limit of cos(x)/factorial(x)
-
Limit of asin(1+x)^2/((1+x)*atan(1+x))
-
Limit of 4*x/(1+x)
-
Identical expressions
- sin(six *pi*x)/sin(pi*x)
- sinus of (6 multiply by Pi multiply by x) divide by sinus of ( Pi multiply by x)
- sinus of (six multiply by Pi multiply by x) divide by sinus of ( Pi multiply by x)
- sin(6pix)/sin(pix)
- sin6pix/sinpix
- sin(6*pi*x) divide by sin(pi*x)
Limit of the function sin(6*pi*x)/sin(pi*x)
The solution
You have entered
[src]
/sin(6*pi*x)\ lim |-----------| x->1+\ sin(pi*x) /
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
Limit(sin((6*pi)*x)/sin(pi*x), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+} \sin{\left(6 \pi x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \sin{\left(\pi x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \sin{\left(6 \pi x \right)}}{\frac{d}{d x} \sin{\left(\pi x \right)}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{6 \cos{\left(6 \pi x \right)}}{\cos{\left(\pi x \right)}}\right)$$
=
$$\lim_{x \to 1^+} -6$$
=
$$\lim_{x \to 1^+} -6$$
=
$$-6$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 1^+} \sin{\left(6 \pi x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+} \sin{\left(\pi x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \sin{\left(6 \pi x \right)}}{\frac{d}{d x} \sin{\left(\pi x \right)}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{6 \cos{\left(6 \pi x \right)}}{\cos{\left(\pi x \right)}}\right)$$
=
$$\lim_{x \to 1^+} -6$$
=
$$\lim_{x \to 1^+} -6$$
=
$$-6$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = -6$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = -6$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = 6$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = 6$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = -6$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = 6$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right) = 6$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
More at x→-oo
One‐sided limits
[src]
/sin(6*pi*x)\ lim |-----------| x->1+\ sin(pi*x) /
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
-6
$$-6$$
= -6.0
/sin(6*pi*x)\ lim |-----------| x->1-\ sin(pi*x) /
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(6 \pi x \right)}}{\sin{\left(\pi x \right)}}\right)$$
-6
$$-6$$
= -6.0
= -6.0
The graph