Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of x/sin(x)
-
Limit of tan(5*x)/sin(2*x)
-
Limit of -x
-
Limit of x/sqrt(1+x^2)
- Derivative of:
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x/sin(x)
- Graphing y =:
-
x/sin(x)
- Integral of d{x}:
- x/sin(x)
-
Identical expressions
- x/sin(x)
- x divide by sinus of (x)
- x/sinx
- x divide by sin(x)
-
Similar expressions
- sin(2*x)/sin(x)^2
- asin(x)/sin(x)
- ((x-sin(x))/sin(x))^(sin(x)/(x-sin(x)))
- sin(3*x)/sin(x)
- (e^(a*x)-e^(b*x))/sin(x)
- x/sinx
Limit of the function x/sin(x)
The solution
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/ x \ lim |------| x->0+\sin(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
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = \lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$\left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{-1}$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = \lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$\left(\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)\right)^{-1}$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$1$$
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
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\cos{\left(x \right)}}$$
=
$$1$$
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
One‐sided limits
[src]
/ x \ lim |------| x->0+\sin(x)/
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1
/ x \ lim |------| x->0-\sin(x)/
$$\lim_{x \to 0^-}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1
= 1
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x}{\sin{\left(x \right)}}\right) = \frac{1}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = \frac{1}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x}{\sin{\left(x \right)}}\right) = \frac{1}{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{\sin{\left(x \right)}}\right) = \frac{1}{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right)$$
More at x→-oo
The graph