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x/sqrt(1-x^2)
Limit of the function/ x/sqrt(1-x^2)

Limit of the function x/sqrt(1-x^2)

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You have entered [src] 
     /     x     \
 lim |-----------|
x->oo|   ________|
     |  /      2 |
     \\/  1 - x  /
lim⁡x→∞(x1−x2)\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
oo/oo*i,

i.e. limit for the numerator is
lim⁡x→∞x=∞\lim_{x \to \infty} x = \infty
and limit for the denominator is
lim⁡x→∞1−x2=∞i\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→∞(x1−x2)\lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right)
=
lim⁡x→∞(ddxxddx1−x2)\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \sqrt{1 - x^{2}}}\right)
=
lim⁡x→∞(−1−x2x)\lim_{x \to \infty}\left(- \frac{\sqrt{1 - x^{2}}}{x}\right)
=
lim⁡x→∞(−1−x2x)\lim_{x \to \infty}\left(- \frac{\sqrt{1 - x^{2}}}{x}\right)
=
−i- 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
02468-8-6-4-2-10105-5
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(x1−x2)=−i\lim_{x \to \infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = - i
lim⁡x→0−(x1−x2)=0\lim_{x \to 0^-}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = 0
More at x→0 from the left
lim⁡x→0+(x1−x2)=0\lim_{x \to 0^+}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = 0
More at x→0 from the right
lim⁡x→1−(x1−x2)=∞\lim_{x \to 1^-}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = \infty
More at x→1 from the left
lim⁡x→1+(x1−x2)=−∞i\lim_{x \to 1^+}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = - \infty i
More at x→1 from the right
lim⁡x→−∞(x1−x2)=i\lim_{x \to -\infty}\left(\frac{x}{\sqrt{1 - x^{2}}}\right) = i
More at x→-oo
Rapid solution [src] 
-I
−i- i
Expand and simplify
The graph
Limit of the function x/sqrt(1-x^2)
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