Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (5+x-3*x^2)/(4-x+2*x^2)
-
Limit of (4-x^2)/(3-x^2)
-
Limit of (3+2*x)/(1-5*x)
-
Limit of (1-2*cos(x))/sin(3*x)
- Integral of d{x}:
- log(1-x^2)/x^2
-
Identical expressions
- log(one -x^ two)/x^ two
- logarithm of (1 minus x squared ) divide by x squared
- logarithm of (one minus x to the power of two) divide by x to the power of two
- log(1-x2)/x2
- log1-x2/x2
- log(1-x²)/x²
- log(1-x to the power of 2)/x to the power of 2
- log1-x^2/x^2
- log(1-x^2) divide by x^2
-
Similar expressions
- 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)
You entered:
log(1-x^2)/x^2
What you mean?
Limit of the function log(1-x^2)/x^2
The solution
You have entered
[src]
/ / 2\\
|log\1 - x /|
lim |-----------|
x->0+| 2 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)$$
Limit(log(1 - x^2)/(x^2), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \log{\left(1 - x^{2} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(1 - x^{2} \right)}}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{1 - x^{2}}\right)$$
=
$$\lim_{x \to 0^+} -1$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+} -1$$
=
$$\lim_{x \to 0^+} -1$$
=
$$-1$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \log{\left(1 - x^{2} \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x^{2} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(1 - x^{2} \right)}}{\frac{d}{d x} x^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{1 - x^{2}}\right)$$
=
$$\lim_{x \to 0^+} -1$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- 2 x\right)}{\frac{d}{d x} 2 x}\right)$$
=
$$\lim_{x \to 0^+} -1$$
=
$$\lim_{x \to 0^+} -1$$
=
$$-1$$
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
One‐sided limits
[src]
/ / 2\\
|log\1 - x /|
lim |-----------|
x->0+| 2 |
\ x /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)$$
-1
$$-1$$
= -1
/ / 2\\
|log\1 - x /|
lim |-----------|
x->0-| 2 |
\ x /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right)$$
-1
$$-1$$
= -1
= -1
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -\infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -\infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\right) = 0$$
More at x→-oo
The graph