Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (4+x^2)/(-6+2*x)
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Limit of ((1+x)/(1+2*x))^x
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Limit of (n/(1+n))^(5+3*n)
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Limit of (9^x-8^x)/asin(3*x)
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Identical expressions
- sqrt(x*sin(x))/x^ two
- square root of (x multiply by sinus of (x)) divide by x squared
- square root of (x multiply by sinus of (x)) divide by x to the power of two
- √(x*sin(x))/x^2
- sqrt(x*sin(x))/x2
- sqrtx*sinx/x2
- sqrt(x*sin(x))/x²
- sqrt(x*sin(x))/x to the power of 2
- sqrt(xsin(x))/x^2
- sqrt(xsin(x))/x2
- sqrtxsinx/x2
- sqrtxsinx/x^2
- sqrt(x*sin(x)) divide by x^2
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Similar expressions
- sqrt(x*sinx)/x^2
Limit of the function sqrt(x*sin(x))/x^2
The solution
You have entered
[src]
/ __________\
|\/ x*sin(x) |
lim |------------|
x->0+| 2 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
Limit(sqrt(x*sin(x))/x^2, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sqrt{x \sin{\left(x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sqrt{x \sin{\left(x \right)}}}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{2 x^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{\frac{d}{d x} \frac{2 x^{2} \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{x \sin{\left(x \right)}}{2} + \cos{\left(x \right)}}{\frac{x^{2} \cos{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}} + \frac{3 x \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{x \sin{\left(x \right)}}{2} + \cos{\left(x \right)}}{\frac{x^{2} \cos{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}} + \frac{3 x \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sqrt{x \sin{\left(x \right)}} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sqrt{x \sin{\left(x \right)}}}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{2 x^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2}\right)}{\frac{d}{d x} \frac{2 x^{2} \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{x \sin{\left(x \right)}}{2} + \cos{\left(x \right)}}{\frac{x^{2} \cos{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}} + \frac{3 x \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{x \sin{\left(x \right)}}{2} + \cos{\left(x \right)}}{\frac{x^{2} \cos{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}} + \frac{3 x \sin{\left(x \right)}}{\sqrt{x \sin{\left(x \right)}}}}\right)$$
=
$$\infty$$
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
One‐sided limits
[src]
/ __________\
|\/ x*sin(x) |
lim |------------|
x->0+| 2 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
oo
$$\infty$$
= 150.999448123822
/ __________\
|\/ x*sin(x) |
lim |------------|
x->0-| 2 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
oo
$$\infty$$
= 150.999448123822
= 150.999448123822
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \sqrt{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \sqrt{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \sqrt{\sin{\left(1 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right) = \sqrt{\sin{\left(1 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{x \sin{\left(x \right)}}}{x^{2}}\right)$$
More at x→-oo
The graph