Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (5-4*x+3*x^2)/(1-x+2*x^2)
-
Limit of ((1+2*x)/(-1+x))^(4*x)
-
Limit of (4+x^2-5*x)/(-16+x^2)
-
Limit of sinh(x)/x
- Derivative of:
-
x/sqrt(1-x^2)
- x/sqrt(1-x^2)
- Integral of d{x}:
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x/sqrt(1-x^2)
-
Identical expressions
- x/sqrt(one -x^ two)
- x divide by square root of (1 minus x squared )
- x divide by square root of (one minus x to the power of two)
- x/√(1-x^2)
- x/sqrt(1-x2)
- x/sqrt1-x2
- x/sqrt(1-x²)
- x/sqrt(1-x to the power of 2)
- x/sqrt1-x^2
- x divide by sqrt(1-x^2)
-
Similar expressions
- x/sqrt(1+x^2)
- asin(x)/sqrt(1-x^2)
Limit of the function x/sqrt(1-x^2)
The solution
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[src]
/ x \
lim |-----------|
x->oo| ________|
| / 2 |
\\/ 1 - x /
$$\lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right)$$
Limit(x/sqrt(1 - x^2), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} x = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} \sqrt{1 - x^{2}} = \infty i$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sqrt{1 - x^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\sqrt{1 - x^{2}}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\sqrt{1 - x^{2}}}{x}\right)$$
=
$$- i$$
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)
oo/oo*i,
i.e. limit for the numerator is
$$\lim_{x \to \infty} x = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} \sqrt{1 - x^{2}} = \infty i$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sqrt{1 - x^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\sqrt{1 - x^{2}}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\sqrt{1 - x^{2}}}{x}\right)$$
=
$$- i$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = - i$$
$$\lim_{x \to 0^-}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = - \infty i$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = i$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = \infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = - \infty i$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = i$$
More at x→-oo
The graph