Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-5+7*n)/(2+n)
-
Limit of (-2+x)/(-4+x)
-
Limit of (-16+x^2-6*x)/(-2+x+x^2)
-
Limit of (2+x^2-5*x)/(5+x^2+3*x^3)
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Identical expressions
- log(one -x^ three)/x
- logarithm of (1 minus x cubed ) divide by x
- logarithm of (one minus x to the power of three) divide by x
- log(1-x3)/x
- log1-x3/x
- log(1-x³)/x
- log(1-x to the power of 3)/x
- log1-x^3/x
- log(1-x^3) divide by x
-
Similar expressions
- log(1+x^3)/x
-
What you mean?
- log(1 - x^3)/x
- log(1 - x*3)/x
- log(1 - x^3)/x
- log(1 - x^3)/x
You entered:
log(1-x^3)/x
What you mean?
Limit of the function log(1-x^3)/x
The solution
You have entered
[src]
/ / 3\\
|log\1 - x /|
lim |-----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right)$$
Limit(log(1 - x^3)/x, x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \log{\left(- x^{3} + 1 \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(- x^{3} + 1 \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 x^{2}}{- x^{3} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(- 3 x^{2}\right)$$
=
$$\lim_{x \to 0^+}\left(- 3 x^{2}\right)$$
=
$$0$$
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
$$\lim_{x \to 0^+} \log{\left(- x^{3} + 1 \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \log{\left(- x^{3} + 1 \right)}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{3 x^{2}}{- x^{3} + 1}\right)$$
=
$$\lim_{x \to 0^+}\left(- 3 x^{2}\right)$$
=
$$\lim_{x \to 0^+}\left(- 3 x^{2}\right)$$
=
$$0$$
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]
/ / 3\\
|log\1 - x /|
lim |-----------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right)$$
0
$$0$$
= 8.98284195579066e-31
/ / 3\\
|log\1 - x /|
lim |-----------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right)$$
0
$$0$$
= 8.35837439466625e-30
= 8.35837439466625e-30
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = -\infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = -\infty$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = -\infty$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(- x^{3} + 1 \right)}}{x}\right) = 0$$
More at x→-oo
The graph