Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-2*x^2+4*x^3+5*x)/(3*x^2+7*x)
-
Limit of (-cos(5*x)+cos(3*x))/x^2
-
Limit of (-1+cos(7*x))/(-1+cos(3*x))
-
Limit of (16+x^2+10*x)/(-6+x^2-x)
- Integral of d{x}:
- asin(x)/x
-
Identical expressions
- asin(x)/x
- arc sinus of e of (x) divide by x
- asinx/x
- asin(x) divide by x
-
Similar expressions
- asin(x)/(x-sin(x))
- a^x-a^sin(x)/x^3
- x*asin(x)/(x-sin(x))
- arcsin(x)/x
- arcsinx/x
Limit of the function asin(x)/x
The solution
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[src]
/asin(x)\ lim |-------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
Limit(asin(x)/x, x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
Do replacement
$$u = \operatorname{asin}{\left(x \right)}$$
$$x = \sin{\left(u \right)}$$
we get
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{\operatorname{asin}{\left(\frac{\sin{\left(u \right)}}{1} \right)}}{1^{-1} \sin{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\operatorname{asin}{\left(\sin{\left(u \right)} \right)}}{\sin{\left(u \right)}}\right) = \lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+} \frac{1}{\frac{1}{u} \sin{\left(u \right)}}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
Do replacement
$$u = \operatorname{asin}{\left(x \right)}$$
$$x = \sin{\left(u \right)}$$
we get
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{\operatorname{asin}{\left(\frac{\sin{\left(u \right)}}{1} \right)}}{1^{-1} \sin{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+}\left(\frac{\operatorname{asin}{\left(\sin{\left(u \right)} \right)}}{\sin{\left(u \right)}}\right) = \lim_{u \to 0^+}\left(\frac{u}{\sin{\left(u \right)}}\right)$$
=
$$\lim_{u \to 0^+} \frac{1}{\frac{1}{u} \sin{\left(u \right)}}$$
/sin(u)\
= 1 / ( lim |------| )
u->0+\ u / 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:
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = 1$$
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
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{\sqrt{1 - x^{2}}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\sqrt{1 - x^{2}}}$$
=
$$1$$
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
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(x \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{\sqrt{1 - x^{2}}}$$
=
$$\lim_{x \to 0^+} \frac{1}{\sqrt{1 - x^{2}}}$$
=
$$1$$
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
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/asin(x)\ lim |-------| x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
/asin(x)\ lim |-------| x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
1
$$1$$
= 1.0
= 1.0
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = \frac{\pi}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = \frac{\pi}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = \frac{\pi}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right) = \frac{\pi}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(x \right)}}{x}\right)$$
More at x→-oo
The graph