Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-12+x^2+4*x)/(-4+x^2)
-
Limit of (-10+x^2+3*x)/(16+x^2-10*x)
-
Limit of (6+x^2+2*x)/(-1+3*x^2+7*x)
-
Limit of ((-1+x^2)/(2+x^2))^(1+x^2)
- Derivative of:
-
log(x)/(1-x)
- Integral of d{x}:
- log(x)/(1-x)
-
Identical expressions
- log(x)/(one -x)
- logarithm of (x) divide by (1 minus x)
- logarithm of (x) divide by (one minus x)
- logx/1-x
- log(x) divide by (1-x)
-
Similar expressions
- log(x)/(1+x)
- sqrt(x)*log(x)/(1-x^2)
- cos(pi*x/2)*log(x)/(1-x^2)
Limit of the function log(x)/(1-x)
The solution
You have entered
[src]
/log(x)\ lim |------| x->1+\1 - x /
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right)$$
Limit(log(x)/(1 - x), 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
$$\lim_{x \to 1^+}\left(1 - x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} \left(1 - x\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{1}{x}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{1}{x}\right)$$
=
$$-1$$
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
$$\lim_{x \to 1^+}\left(1 - x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} \left(1 - x\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{1}{x}\right)$$
=
$$\lim_{x \to 1^+}\left(- \frac{1}{x}\right)$$
=
$$-1$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = -1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = 0$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = -1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{1 - x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/log(x)\ lim |------| x->1+\1 - x /
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{1 - x}\right)$$
-1
$$-1$$
= -1.0
/log(x)\ lim |------| x->1-\1 - x /
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}}{1 - x}\right)$$
-1
$$-1$$
= -1.0
= -1.0
The graph