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

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

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     /      2\
     |-1 + x |
 lim |-------|
x->1+\ 1 - x /
lim⁡x→1+(x2−11−x)\lim_{x \to 1^+}\left(\frac{x^{2} - 1}{1 - x}\right) 
Limit((-1 + x^2)/(1 - x), x, 1)
Detail solution
Let's take the limit
lim⁡x→1+(x2−11−x)\lim_{x \to 1^+}\left(\frac{x^{2} - 1}{1 - x}\right)
transform
lim⁡x→1+(x2−11−x)\lim_{x \to 1^+}\left(\frac{x^{2} - 1}{1 - x}\right)
=
lim⁡x→1+((x−1)(x+1)1−x)\lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(x + 1\right)}{1 - x}\right)
=
lim⁡x→1+(−x−1)=\lim_{x \to 1^+}\left(- x - 1\right) =
−1−1=-1 - 1 = 
= -2

The final answer:
lim⁡x→1+(x2−11−x)=−2\lim_{x \to 1^+}\left(\frac{x^{2} - 1}{1 - x}\right) = -2
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→1+(x2−1)=0\lim_{x \to 1^+}\left(x^{2} - 1\right) = 0
and limit for the denominator is
lim⁡x→1+(1−x)=0\lim_{x \to 1^+}\left(1 - x\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→1+(x2−11−x)\lim_{x \to 1^+}\left(\frac{x^{2} - 1}{1 - x}\right)
=
lim⁡x→1+(ddx(x2−1)ddx(1−x))\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(x^{2} - 1\right)}{\frac{d}{d x} \left(1 - x\right)}\right)
=
lim⁡x→1+(−2x)\lim_{x \to 1^+}\left(- 2 x\right)
=
lim⁡x→1+−2\lim_{x \to 1^+} -2
=
lim⁡x→1+−2\lim_{x \to 1^+} -2
=
−2-2
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
-2.0-1.5-1.0-0.52.00.00.51.01.55-5
Plot the graph
Rapid solution [src] 
-2
−2-2
Expand and simplify
One‐sided limits [src] 
     /      2\
     |-1 + x |
 lim |-------|
x->1+\ 1 - x /
lim⁡x→1+(x2−11−x)\lim_{x \to 1^+}\left(\frac{x^{2} - 1}{1 - x}\right) 
-2
−2-2 
= -2.0
     /      2\
     |-1 + x |
 lim |-------|
x->1-\ 1 - x /
lim⁡x→1−(x2−11−x)\lim_{x \to 1^-}\left(\frac{x^{2} - 1}{1 - x}\right) 
-2
−2-2 
= -2.0
= -2.0
Other limits x→0, -oo, +oo, 1
lim⁡x→1−(x2−11−x)=−2\lim_{x \to 1^-}\left(\frac{x^{2} - 1}{1 - x}\right) = -2
More at x→1 from the left
lim⁡x→1+(x2−11−x)=−2\lim_{x \to 1^+}\left(\frac{x^{2} - 1}{1 - x}\right) = -2
lim⁡x→∞(x2−11−x)=−∞\lim_{x \to \infty}\left(\frac{x^{2} - 1}{1 - x}\right) = -\infty
More at x→oo
lim⁡x→0−(x2−11−x)=−1\lim_{x \to 0^-}\left(\frac{x^{2} - 1}{1 - x}\right) = -1
More at x→0 from the left
lim⁡x→0+(x2−11−x)=−1\lim_{x \to 0^+}\left(\frac{x^{2} - 1}{1 - x}\right) = -1
More at x→0 from the right
lim⁡x→−∞(x2−11−x)=∞\lim_{x \to -\infty}\left(\frac{x^{2} - 1}{1 - x}\right) = \infty
More at x→-oo
Numerical answer [src] 
-2.0
-2.0
The graph
Limit of the function (-1+x^2)/(1-x)
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