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

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log(1-x^2)/x^2

What you mean?

log(1 - x^2)/(x^2)
 
log(1 - x^2)*2/x
 
log(1 - x^2)/(x^2)
 
log(1 - x^2)*2/x
 
log(1 - x^2)/(x^2)
 
log(1 - x^2)*2/x
 
log(1 - x^2)/(x^2)

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

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The solution

You have entered [src] 
     /   /     2\\
     |log\1 - x /|
 lim |-----------|
x->0+|      2    |
     \     x     /
lim⁡x→0+(log⁡(1−x2)x2)\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) 
Limit(log(1 - x^2)/(x^2), x, 0)
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+log⁡(1−x2)=0\lim_{x \to 0^+} \log{\left(1 - x^{2} \right)} = 0
and limit for the denominator is
lim⁡x→0+x2=0\lim_{x \to 0^+} x^{2} = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(log⁡(1−x2)x2)\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(log⁡(1−x2)x2)\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)
=
lim⁡x→0+(ddxlog⁡(1−x2)ddxx2)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(1 - x^{2} \right)}}{\frac{d}{d x} x^{2}}\right)
=
lim⁡x→0+(−11−x2)\lim_{x \to 0^+}\left(- \frac{1}{1 - x^{2}}\right)
=
lim⁡x→0+−1\lim_{x \to 0^+} -1
=
lim⁡x→0+(ddx(−2x)ddx2x)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x\right)}{\frac{d}{d x} 2 x}\right)
=
lim⁡x→0+−1\lim_{x \to 0^+} -1
=
lim⁡x→0+−1\lim_{x \to 0^+} -1
=
−1-1
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 2 time(s)
The graph
02468-8-6-4-2-10100-3
Plot the graph
Rapid solution [src] 
-1
−1-1
Expand and simplify
One‐sided limits [src] 
     /   /     2\\
     |log\1 - x /|
 lim |-----------|
x->0+|      2    |
     \     x     /
lim⁡x→0+(log⁡(1−x2)x2)\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) 
-1
−1-1 
= -1
     /   /     2\\
     |log\1 - x /|
 lim |-----------|
x->0-|      2    |
     \     x     /
lim⁡x→0−(log⁡(1−x2)x2)\lim_{x \to 0^-}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) 
-1
−1-1 
= -1
= -1
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(log⁡(1−x2)x2)=−1\lim_{x \to 0^-}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -1
More at x→0 from the left
lim⁡x→0+(log⁡(1−x2)x2)=−1\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -1
lim⁡x→∞(log⁡(1−x2)x2)=0\lim_{x \to \infty}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = 0
More at x→oo
lim⁡x→1−(log⁡(1−x2)x2)=−∞\lim_{x \to 1^-}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -\infty
More at x→1 from the left
lim⁡x→1+(log⁡(1−x2)x2)=−∞\lim_{x \to 1^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -\infty
More at x→1 from the right
lim⁡x→−∞(log⁡(1−x2)x2)=0\lim_{x \to -\infty}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = 0
More at x→-oo
Numerical answer [src] 
-1.0
-1.0
The graph
Limit of the function log(1-x^2)/x^2
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