Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-3+x^2-2*x)/(-3+x)
-
Limit of 5^x-cos(x)
-
Limit of (x/2)^(1/(-2+x))
-
Limit of (1-3/x)^x
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Identical expressions
- asin(x)/sin(x)
- arc sinus of e of (x) divide by sinus of (x)
- asinx/sinx
- asin(x) divide by sin(x)
-
Similar expressions
- arcsin(x)/sin(x)
- arcsinx/sinx
Limit of the function asin(x)/sin(x)
The solution
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[src]
/asin(x)\ lim |-------| x->0+\ sin(x)/
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
Limit(asin(x)/sin(x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \operatorname{asin}{\left(x \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(x \right)}}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\sqrt{1 - x^{2}} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\sqrt{1 - x^{2}} \cos{\left(x \right)}}\right)$$
=
$$1$$
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
$$\lim_{x \to 0^+} \operatorname{asin}{\left(x \right)} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(x \right)}}{\frac{d}{d x} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\sqrt{1 - x^{2}} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{\sqrt{1 - x^{2}} \cos{\left(x \right)}}\right)$$
=
$$1$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right) = \frac{\pi}{2 \sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right) = \frac{\pi}{2 \sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right) = \frac{\pi}{2 \sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right) = \frac{\pi}{2 \sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
More at x→-oo
One‐sided limits
[src]
/asin(x)\ lim |-------| x->0+\ sin(x)/
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1
/asin(x)\ lim |-------| x->0-\ sin(x)/
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{\sin{\left(x \right)}}\right)$$
1
$$1$$
= 1
= 1
The graph