Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of e^(x/2)
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Limit of (1+x)^3-(-1+x)^3/(1+x^2)
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Limit of (-24+x^2+2*x)/(-4+x)
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Limit of (1+x^2-2*x)/(1+x^3)
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Identical expressions
- asin(five *x)/(four *sin(x)^ two)
- arc sinus of e of (5 multiply by x) divide by (4 multiply by sinus of (x) squared )
- arc sinus of e of (five multiply by x) divide by (four multiply by sinus of (x) to the power of two)
- asin(5*x)/(4*sin(x)2)
- asin5*x/4*sinx2
- asin(5*x)/(4*sin(x)²)
- asin(5*x)/(4*sin(x) to the power of 2)
- asin(5x)/(4sin(x)^2)
- asin(5x)/(4sin(x)2)
- asin5x/4sinx2
- asin5x/4sinx^2
- asin(5*x) divide by (4*sin(x)^2)
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Similar expressions
- arcsin(5*x)/(4*sin(x)^2)
- asin(5*x)/(4*sinx^2)
Limit of the function asin(5*x)/(4*sin(x)^2)
The solution
You have entered
[src]
/asin(5*x)\
lim |---------|
x->0+| 2 |
\4*sin (x)/
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
Limit(asin(5*x)/((4*sin(x)^2)), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \operatorname{asin}{\left(5 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(4 \sin^{2}{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(5 x \right)}}{\frac{d}{d x} 4 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 \sqrt{1 - 25 x^{2}} \sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 \sin{\left(x \right)}}\right)$$
=
$$\infty$$
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
$$\lim_{x \to 0^+} \operatorname{asin}{\left(5 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(4 \sin^{2}{\left(x \right)}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \operatorname{asin}{\left(5 x \right)}}{\frac{d}{d x} 4 \sin^{2}{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 \sqrt{1 - 25 x^{2}} \sin{\left(x \right)} \cos{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{5}{8 \sin{\left(x \right)}}\right)$$
=
$$\infty$$
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(5*x)\
lim |---------|
x->0+| 2 |
\4*sin (x)/
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
oo
$$\infty$$
= 188.787269213921
/asin(5*x)\
lim |---------|
x->0-| 2 |
\4*sin (x)/
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
-oo
$$-\infty$$
= -188.787269213921
= -188.787269213921
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{4 \sin^{2}{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{4 \sin^{2}{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{4 \sin^{2}{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right) = \frac{\operatorname{asin}{\left(5 \right)}}{4 \sin^{2}{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(5 x \right)}}{4 \sin^{2}{\left(x \right)}}\right)$$
More at x→-oo
The graph