Mister Exam

Limit of the function x/sin(x)

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The solution

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     /  x   \
 lim |------|
x->0+\sin(x)/
limx0+(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
limx0+(xsin(x))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)
limx0+(xsin(x))=limu0+(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)
=
limu0+(usin(u))\lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)
=
(limu0+(sin(u)u))1\left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{-1}
The limit
limu0+(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:
limx0+(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
limx0+x=0\lim_{x \to 0^+} x = 0
and limit for the denominator is
limx0+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.
limx0+(xsin(x))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)
=
limx0+(ddxxddxsin(x))\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sin{\left(x \right)}}\right)
=
limx0+1cos(x)\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}
=
limx0+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
Rapid solution [src]
1
11
One‐sided limits [src]
     /  x   \
 lim |------|
x->0+\sin(x)/
limx0+(xsin(x))\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)
1
11
= 1
     /  x   \
 lim |------|
x->0-\sin(x)/
limx0(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
limx0(xsin(x))=1\lim_{x \to 0^-}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1
More at x→0 from the left
limx0+(xsin(x))=1\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1
limx(xsin(x))\lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
More at x→oo
limx1(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
limx1+(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
limx(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)