Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of sin(2*x)/(5*x)
-
Limit of ((-5+2*x)/(3+2*x))^(7*x)
-
Limit of (pi/2-atan(x))^(1/x)
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Limit of log(x)/(-1+x^2)
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Identical expressions
- log(x)/(- one +x^ two)
- logarithm of (x) divide by ( minus 1 plus x squared )
- logarithm of (x) divide by ( minus one plus x to the power of two)
- log(x)/(-1+x2)
- logx/-1+x2
- log(x)/(-1+x²)
- log(x)/(-1+x to the power of 2)
- logx/-1+x^2
- log(x) divide by (-1+x^2)
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Similar expressions
- log(x)/(-1-x^2)
- log(x)/(1+x^2)
Limit of the function log(x)/(-1+x^2)
The solution
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[src]
/ log(x)\
lim |-------|
x->1+| 2|
\-1 + x /
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right)$$
Limit(log(x)/(-1 + x^2), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+} \log{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x^{2} - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} \left(x^{2} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{2 x^{2}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{2 x}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{2 x}\right)$$
=
$$\frac{1}{2}$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 1^+} \log{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x^{2} - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} \left(x^{2} - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{2 x^{2}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{2 x}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{1}{2 x}\right)$$
=
$$\frac{1}{2}$$
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
One‐sided limits
[src]
/ log(x)\
lim |-------|
x->1+| 2|
\-1 + x /
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
/ log(x)\
lim |-------|
x->1-| 2|
\-1 + x /
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
= 0.5
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = \frac{1}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = 0$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x^{2} - 1}\right) = 0$$
More at x→-oo
The graph