Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-64+4^x)/(-3+x)
-
Limit of (3-x^2+5*x)/(4*x^7+81*x)
-
Limit of (-2+2*x^3+7*x)/(-4-x+3*x^3)
-
Limit of (1-sin(x))/cos(x)
- Graphing y =:
- x-sqrt(x^2-x)
- Derivative of:
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x-sqrt(x^2-x)
-
Identical expressions
- x-sqrt(x^ two -x)
- x minus square root of (x squared minus x)
- x minus square root of (x to the power of two minus x)
- x-√(x^2-x)
- x-sqrt(x2-x)
- x-sqrtx2-x
- x-sqrt(x²-x)
- x-sqrt(x to the power of 2-x)
- x-sqrtx^2-x
-
Similar expressions
- x+sqrt(x^2-x)
- x-sqrt(x^2+x)
Limit of the function x-sqrt(x^2-x)
The solution
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lim \x - \/ x - x /
x->oo
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right)$$
Limit(x - sqrt(x^2 - x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right)$$
Let's eliminate indeterminateness oo - oo
Multiply and divide by
$$x + \sqrt{x^{2} - x}$$
then
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(x - \sqrt{x^{2} - x}\right) \left(x + \sqrt{x^{2} - x}\right)}{x + \sqrt{x^{2} - x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{x^{2} - \left(\sqrt{x^{2} - x}\right)^{2}}{x + \sqrt{x^{2} - x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{x}{x + \sqrt{x^{2} - x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{x}{x + \sqrt{x^{2} - x}}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty} \frac{1}{1 + \frac{\sqrt{x^{2} - x}}{x}}$$ =
$$\lim_{x \to \infty} \frac{1}{\sqrt{\frac{x^{2} - x}{x^{2}}} + 1}$$ =
$$\lim_{x \to \infty} \frac{1}{\sqrt{1 - \frac{1}{x}} + 1}$$
Do replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{\sqrt{1 - \frac{1}{x}} + 1}$$ =
$$\lim_{u \to 0^+} \frac{1}{\sqrt{1 - u} + 1}$$ =
= $$\frac{1}{1 + \sqrt{1 - 0}} = \frac{1}{2}$$
The final answer:
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right)$$
Let's eliminate indeterminateness oo - oo
Multiply and divide by
$$x + \sqrt{x^{2} - x}$$
then
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(x - \sqrt{x^{2} - x}\right) \left(x + \sqrt{x^{2} - x}\right)}{x + \sqrt{x^{2} - x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{x^{2} - \left(\sqrt{x^{2} - x}\right)^{2}}{x + \sqrt{x^{2} - x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{x}{x + \sqrt{x^{2} - x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{x}{x + \sqrt{x^{2} - x}}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty} \frac{1}{1 + \frac{\sqrt{x^{2} - x}}{x}}$$ =
$$\lim_{x \to \infty} \frac{1}{\sqrt{\frac{x^{2} - x}{x^{2}}} + 1}$$ =
$$\lim_{x \to \infty} \frac{1}{\sqrt{1 - \frac{1}{x}} + 1}$$
Do replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{\sqrt{1 - \frac{1}{x}} + 1}$$ =
$$\lim_{u \to 0^+} \frac{1}{\sqrt{1 - u} + 1}$$ =
= $$\frac{1}{1 + \sqrt{1 - 0}} = \frac{1}{2}$$
The final answer:
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right) = \frac{1}{2}$$
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
$$\lim_{x \to \infty}\left(x - \sqrt{x^{2} - x}\right) = \frac{1}{2}$$
$$\lim_{x \to 0^-}\left(x - \sqrt{x^{2} - x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x - \sqrt{x^{2} - x}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x - \sqrt{x^{2} - x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x - \sqrt{x^{2} - x}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x - \sqrt{x^{2} - x}\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x - \sqrt{x^{2} - x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x - \sqrt{x^{2} - x}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x - \sqrt{x^{2} - x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x - \sqrt{x^{2} - x}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x - \sqrt{x^{2} - x}\right) = -\infty$$
More at x→-oo
The graph