Mister Exam
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How to use it?
- Limit of the function:
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Limit of (x/(1+x))^(1/3)
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Limit of (3+sqrt(x)-sqrt(3))/x
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Limit of sin(-1+x)/(-1+x)^2
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Limit of 1+tan(w)^2-1/tan(w)^2
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Identical expressions
- (sqrt(five +x)-sqrt(two)*sqrt(x))/(sqrt(three +x)-sqrt(thirteen -x))
- ( square root of (5 plus x) minus square root of (2) multiply by square root of (x)) divide by ( square root of (3 plus x) minus square root of (13 minus x))
- ( square root of (five plus x) minus square root of (two) multiply by square root of (x)) divide by ( square root of (three plus x) minus square root of (thirteen minus x))
- (√(5+x)-√(2)*√(x))/(√(3+x)-√(13-x))
- (sqrt(5+x)-sqrt(2)sqrt(x))/(sqrt(3+x)-sqrt(13-x))
- sqrt5+x-sqrt2sqrtx/sqrt3+x-sqrt13-x
- (sqrt(5+x)-sqrt(2)*sqrt(x)) divide by (sqrt(3+x)-sqrt(13-x))
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Similar expressions
- (sqrt(5+x)-sqrt(2)*sqrt(x))/(sqrt(3+x)+sqrt(13-x))
- (sqrt(5+x)-sqrt(2)*sqrt(x))/(sqrt(3-x)-sqrt(13-x))
- (sqrt(5+x)+sqrt(2)*sqrt(x))/(sqrt(3+x)-sqrt(13-x))
- (sqrt(5-x)-sqrt(2)*sqrt(x))/(sqrt(3+x)-sqrt(13-x))
- (sqrt(5+x)-sqrt(2)*sqrt(x))/(sqrt(3+x)-sqrt(13+x))
Limit of the function (sqrt(5+x)-sqrt(2)*sqrt(x))/(sqrt(3+x)-sqrt(13-x))
The solution
You have entered
[src]
/ _______ ___ ___\
|\/ 5 + x - \/ 2 *\/ x |
lim |-----------------------|
x->5+| _______ ________|
\ \/ 3 + x - \/ 13 - x /
$$\lim_{x \to 5^+}\left(\frac{- \sqrt{2} \sqrt{x} + \sqrt{x + 5}}{- \sqrt{13 - x} + \sqrt{x + 3}}\right)$$
Limit((sqrt(5 + x) - sqrt(2)*sqrt(x))/(sqrt(3 + x) - sqrt(13 - x)), x, 5)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 5^+}\left(- \sqrt{2} \sqrt{x} + \sqrt{x + 5}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 5^+}\left(- \sqrt{13 - x} + \sqrt{x + 3}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 5^+}\left(\frac{- \sqrt{2} \sqrt{x} + \sqrt{x + 5}}{- \sqrt{13 - x} + \sqrt{x + 3}}\right)$$
=
$$\lim_{x \to 5^+}\left(\frac{\frac{d}{d x} \left(- \sqrt{2} \sqrt{x} + \sqrt{x + 5}\right)}{\frac{d}{d x} \left(- \sqrt{13 - x} + \sqrt{x + 3}\right)}\right)$$
=
$$\lim_{x \to 5^+}\left(\frac{\frac{1}{2 \sqrt{x + 5}} - \frac{\sqrt{2}}{2 \sqrt{x}}}{\frac{1}{2 \sqrt{x + 3}} + \frac{1}{2 \sqrt{13 - x}}}\right)$$
=
$$\lim_{x \to 5^+}\left(\frac{\frac{1}{2 \sqrt{x + 5}} - \frac{\sqrt{2}}{2 \sqrt{x}}}{\frac{1}{2 \sqrt{x + 3}} + \frac{1}{2 \sqrt{13 - x}}}\right)$$
=
$$- \frac{\sqrt{5}}{5}$$
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
$$\lim_{x \to 5^+}\left(- \sqrt{2} \sqrt{x} + \sqrt{x + 5}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 5^+}\left(- \sqrt{13 - x} + \sqrt{x + 3}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 5^+}\left(\frac{- \sqrt{2} \sqrt{x} + \sqrt{x + 5}}{- \sqrt{13 - x} + \sqrt{x + 3}}\right)$$
=
$$\lim_{x \to 5^+}\left(\frac{\frac{d}{d x} \left(- \sqrt{2} \sqrt{x} + \sqrt{x + 5}\right)}{\frac{d}{d x} \left(- \sqrt{13 - x} + \sqrt{x + 3}\right)}\right)$$
=
$$\lim_{x \to 5^+}\left(\frac{\frac{1}{2 \sqrt{x + 5}} - \frac{\sqrt{2}}{2 \sqrt{x}}}{\frac{1}{2 \sqrt{x + 3}} + \frac{1}{2 \sqrt{13 - x}}}\right)$$
=
$$\lim_{x \to 5^+}\left(\frac{\frac{1}{2 \sqrt{x + 5}} - \frac{\sqrt{2}}{2 \sqrt{x}}}{\frac{1}{2 \sqrt{x + 3}} + \frac{1}{2 \sqrt{13 - x}}}\right)$$
=
$$- \frac{\sqrt{5}}{5}$$
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]
/ _______ ___ ___\
|\/ 5 + x - \/ 2 *\/ x |
lim |-----------------------|
x->5+| _______ ________|
\ \/ 3 + x - \/ 13 - x /
$$\lim_{x \to 5^+}\left(\frac{- \sqrt{2} \sqrt{x} + \sqrt{x + 5}}{- \sqrt{13 - x} + \sqrt{x + 3}}\right)$$
___ -\/ 5 ------- 5
$$- \frac{\sqrt{5}}{5}$$
= -0.447213595499958
/ _______ ___ ___\
|\/ 5 + x - \/ 2 *\/ x |
lim |-----------------------|
x->5-| _______ ________|
\ \/ 3 + x - \/ 13 - x /
$$\lim_{x \to 5^-}\left(\frac{- \sqrt{2} \sqrt{x} + \sqrt{x + 5}}{- \sqrt{13 - x} + \sqrt{x + 3}}\right)$$
___ -\/ 5 ------- 5
$$- \frac{\sqrt{5}}{5}$$
= -0.447213595499958
= -0.447213595499958
The graph