Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of sin(5*x)/(4*x^2)
-
Limit of sqrt(2+x)-sqrt(x)
-
Limit of (e^x-e^(-x))/(cos(x)*sin(x))
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Limit of (1-cos(2*x))*cot(4*x)
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Identical expressions
- sqrt(two +x)-sqrt(x)
- square root of (2 plus x) minus square root of (x)
- square root of (two plus x) minus square root of (x)
- √(2+x)-√(x)
- sqrt2+x-sqrtx
-
Similar expressions
- sqrt(2-x)-sqrt(x)
- sqrt(2+x)+sqrt(x)
Limit of the function sqrt(2+x)-sqrt(x)
The solution
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/ _______ ___\ lim \\/ 2 + x - \/ x / x->oo
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right)$$
Limit(sqrt(2 + x) - sqrt(x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right)$$
Let's eliminate indeterminateness oo - oo
Multiply and divide by
$$\sqrt{x} + \sqrt{x + 2}$$
then
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \sqrt{x} + \sqrt{x + 2}\right) \left(\sqrt{x} + \sqrt{x + 2}\right)}{\sqrt{x} + \sqrt{x + 2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \left(\sqrt{x}\right)^{2} + \left(\sqrt{x + 2}\right)^{2}}{\sqrt{x} + \sqrt{x + 2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- x + \left(x + 2\right)}{\sqrt{x} + \sqrt{x + 2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} + \sqrt{x + 2}}\right)$$
Let's divide numerator and denominator by sqrt(x):
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(1 + \frac{\sqrt{x + 2}}{\sqrt{x}}\right)}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(\sqrt{\frac{x + 2}{x}} + 1\right)}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(\sqrt{1 + \frac{2}{x}} + 1\right)}\right)$$
Do replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(\sqrt{1 + \frac{2}{x}} + 1\right)}\right)$$ =
$$\lim_{u \to 0^+}\left(\frac{2}{\left(\sqrt{2 u + 1} + 1\right) \sqrt{\frac{1}{u}}}\right)$$ =
= $$\frac{2}{\tilde{\infty} \left(1 + \sqrt{0 \cdot 2 + 1}\right)} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right) = 0$$
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right)$$
Let's eliminate indeterminateness oo - oo
Multiply and divide by
$$\sqrt{x} + \sqrt{x + 2}$$
then
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(- \sqrt{x} + \sqrt{x + 2}\right) \left(\sqrt{x} + \sqrt{x + 2}\right)}{\sqrt{x} + \sqrt{x + 2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \left(\sqrt{x}\right)^{2} + \left(\sqrt{x + 2}\right)^{2}}{\sqrt{x} + \sqrt{x + 2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- x + \left(x + 2\right)}{\sqrt{x} + \sqrt{x + 2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} + \sqrt{x + 2}}\right)$$
Let's divide numerator and denominator by sqrt(x):
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(1 + \frac{\sqrt{x + 2}}{\sqrt{x}}\right)}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(\sqrt{\frac{x + 2}{x}} + 1\right)}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(\sqrt{1 + \frac{2}{x}} + 1\right)}\right)$$
Do replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{2}{\sqrt{x} \left(\sqrt{1 + \frac{2}{x}} + 1\right)}\right)$$ =
$$\lim_{u \to 0^+}\left(\frac{2}{\left(\sqrt{2 u + 1} + 1\right) \sqrt{\frac{1}{u}}}\right)$$ =
= $$\frac{2}{\tilde{\infty} \left(1 + \sqrt{0 \cdot 2 + 1}\right)} = 0$$
The final answer:
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right) = 0$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(- \sqrt{x} + \sqrt{x + 2}\right) = 0$$
$$\lim_{x \to 0^-}\left(- \sqrt{x} + \sqrt{x + 2}\right) = \sqrt{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \sqrt{x} + \sqrt{x + 2}\right) = \sqrt{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(- \sqrt{x} + \sqrt{x + 2}\right) = -1 + \sqrt{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \sqrt{x} + \sqrt{x + 2}\right) = -1 + \sqrt{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \sqrt{x} + \sqrt{x + 2}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(- \sqrt{x} + \sqrt{x + 2}\right) = \sqrt{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(- \sqrt{x} + \sqrt{x + 2}\right) = \sqrt{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(- \sqrt{x} + \sqrt{x + 2}\right) = -1 + \sqrt{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(- \sqrt{x} + \sqrt{x + 2}\right) = -1 + \sqrt{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(- \sqrt{x} + \sqrt{x + 2}\right) = 0$$
More at x→-oo
The graph