Mister Exam

Other calculators:

x/sin(x)
Limit of the function/ x/sin(x)

Limit of the function x/sin(x)

at →
v

For end points:

The graph:

from to

Piecewise:

The solution

You have entered [src] 
     /  x   \
 lim |------|
x->0+\sin(x)/
lim⁡x→0+(xsin⁡(x))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) 
Limit(x/sin(x), x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(xsin⁡(x))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)
lim⁡x→0+(xsin⁡(x))=lim⁡u→0+(usin⁡(u))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = \lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)
=
lim⁡u→0+(usin⁡(u))\lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)
=
(lim⁡u→0+(sin⁡(u)u))−1\left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{-1}
The limit
lim⁡u→0+(sin⁡(u)u)\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)
is first remarkable limit, is equal to 1.

The final answer:
lim⁡x→0+(xsin⁡(x))=1\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+x=0\lim_{x \to 0^+} x = 0
and limit for the denominator is
lim⁡x→0+sin⁡(x)=0\lim_{x \to 0^+} \sin{\left(x \right)} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(xsin⁡(x))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)
=
lim⁡x→0+(ddxxddxsin⁡(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sin{\left(x \right)}}\right)
=
lim⁡x→0+1cos⁡(x)\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}
=
lim⁡x→0+1cos⁡(x)\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}
=
11
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
02468-8-6-4-2-1010-500500
Plot the graph
Rapid solution [src] 
1
11
Expand and simplify
One‐sided limits [src] 
     /  x   \
 lim |------|
x->0+\sin(x)/
lim⁡x→0+(xsin⁡(x))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) 
1
11 
= 1
     /  x   \
 lim |------|
x->0-\sin(x)/
lim⁡x→0−(xsin⁡(x))\lim_{x \to 0^-}\left(\frac{x}{\sin{\left(x \right)}}\right) 
1
11 
= 1
= 1
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(xsin⁡(x))=1\lim_{x \to 0^-}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1
More at x→0 from the left
lim⁡x→0+(xsin⁡(x))=1\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1
lim⁡x→∞(xsin⁡(x))\lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
More at x→oo
lim⁡x→1−(xsin⁡(x))=1sin⁡(1)\lim_{x \to 1^-}\left(\frac{x}{\sin{\left(x \right)}}\right) = \frac{1}{\sin{\left(1 \right)}}
More at x→1 from the left
lim⁡x→1+(xsin⁡(x))=1sin⁡(1)\lim_{x \to 1^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = \frac{1}{\sin{\left(1 \right)}}
More at x→1 from the right
lim⁡x→−∞(xsin⁡(x))\lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
More at x→-oo
Numerical answer [src] 
1.0
1.0
The graph
Limit of the function x/sin(x)
    © Mister Exam – Calculator