Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (x+sqrt(10+x-x^2))/(sqrt(2-7*x)+2*x)
- Limit of (-1+x^m)/(-1+x)
- Limit of x*a^(-x)
-
Limit of -4+x+x^3
- Graphing y =:
- x/log(1-2*x)
-
Identical expressions
- x/log(one - two *x)
- x divide by logarithm of (1 minus 2 multiply by x)
- x divide by logarithm of (one minus two multiply by x)
- x/log(1-2x)
- x/log1-2x
- x divide by log(1-2*x)
-
Similar expressions
- (-1+e^(3*x))/log(1-2*x)
- x*asin(3*x)/log(1-2*x^2)
- (-1+sqrt(1-6*x))*sin(4*x)/log(1-2*x)^2
- asin(6*x)/log(1-2*x)
- x/log(1+2*x)
Limit of the function x/log(1-2*x)
The solution
You have entered
[src]
/ x \ lim |------------| x->0+\log(1 - 2*x)/
$$\lim_{x \to 0^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right)$$
Limit(x/log(1 - 2*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \log{\left(1 - 2 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \log{\left(1 - 2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(x - \frac{1}{2}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$- \frac{1}{2}$$
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
$$\lim_{x \to 0^+} x = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \log{\left(1 - 2 x \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \log{\left(1 - 2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(x - \frac{1}{2}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$- \frac{1}{2}$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = - \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = - \frac{i}{\pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = - \frac{i}{\pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = -\infty$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = - \frac{i}{\pi}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = - \frac{i}{\pi}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right) = -\infty$$
More at x→-oo
One‐sided limits
[src]
/ x \ lim |------------| x->0+\log(1 - 2*x)/
$$\lim_{x \to 0^+}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
/ x \ lim |------------| x->0-\log(1 - 2*x)/
$$\lim_{x \to 0^-}\left(\frac{x}{\log{\left(1 - 2 x \right)}}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
= -0.5
The graph