Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-2+sqrt(x))/(-4+x^2)
-
Limit of (-exp(-x)+exp(x))/(exp(x)+exp(-x))
-
Limit of x+2*x^3+5*x^4-x^2/3
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Limit of (-x^3+2*x+5*x^4)/(1+x^4-8*x^3)
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Identical expressions
- asin(two *x)/(four *x)
- arc sinus of e of (2 multiply by x) divide by (4 multiply by x)
- arc sinus of e of (two multiply by x) divide by (four multiply by x)
- asin(2x)/(4x)
- asin2x/4x
- asin(2*x) divide by (4*x)
-
Similar expressions
- arcsin(2*x)/(4*x)
Limit of the function asin(2*x)/(4*x)
The solution
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/asin(2*x)\ lim |---------| x->0+\ 4*x /
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
Limit(asin(2*x)/((4*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
Do replacement
$$u = \operatorname{asin}{\left(2 x \right)}$$
$$x = \frac{\sin{\left(u \right)}}{2}$$
we get
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{\lim_{u \to 0^+}\left(\frac{\operatorname{asin}{\left(\frac{2 \sin{\left(u \right)}}{2} \right)}}{\frac{1}{2} \sin{\left(u \right)}}\right)}{4}$$
=
$$\frac{\lim_{u \to 0^+}\left(\frac{2 \operatorname{asin}{\left(\sin{\left(u \right)} \right)}}{\sin{\left(u \right)}}\right)}{4} = \frac{\lim_{u \to 0^+}\left(\frac{2 u}{\sin{\left(u \right)}}\right)}{4}$$
=
$$\frac{\lim_{u \to 0^+} \frac{1}{\frac{1}{u} \sin{\left(u \right)}}}{2}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{1}{2}$$
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
Do replacement
$$u = \operatorname{asin}{\left(2 x \right)}$$
$$x = \frac{\sin{\left(u \right)}}{2}$$
we get
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{\lim_{u \to 0^+}\left(\frac{\operatorname{asin}{\left(\frac{2 \sin{\left(u \right)}}{2} \right)}}{\frac{1}{2} \sin{\left(u \right)}}\right)}{4}$$
=
$$\frac{\lim_{u \to 0^+}\left(\frac{2 \operatorname{asin}{\left(\sin{\left(u \right)} \right)}}{\sin{\left(u \right)}}\right)}{4} = \frac{\lim_{u \to 0^+}\left(\frac{2 u}{\sin{\left(u \right)}}\right)}{4}$$
=
$$\frac{\lim_{u \to 0^+} \frac{1}{\frac{1}{u} \sin{\left(u \right)}}}{2}$$
/sin(u)\
= 1/2 / ( 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:
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{1}{2}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \operatorname{asin}{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(4 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(2 x \right)}}{\frac{d}{d x} 4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{2 \sqrt{1 - 4 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{2}$$
=
$$\lim_{x \to 0^+} \frac{1}{2}$$
=
$$\frac{1}{2}$$
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
$$\lim_{x \to 0^+} \operatorname{asin}{\left(2 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(4 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(2 x \right)}}{\frac{d}{d x} 4 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{1}{2 \sqrt{1 - 4 x^{2}}}\right)$$
=
$$\lim_{x \to 0^+} \frac{1}{2}$$
=
$$\lim_{x \to 0^+} \frac{1}{2}$$
=
$$\frac{1}{2}$$
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]
/asin(2*x)\ lim |---------| x->0+\ 4*x /
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
/asin(2*x)\ lim |---------| x->0-\ 4*x /
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
= 0.5
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right) = \frac{\operatorname{asin}{\left(2 \right)}}{4}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(2 x \right)}}{4 x}\right)$$
More at x→-oo
The graph