Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of x^(4/x)
-
Limit of (-6+x^2-x)/(-4+x^2)
-
Limit of (-6+x^2-x)/(-21+x+2*x^2)
-
Limit of e^(3-x)*(-2+x)
- Integral of d{x}:
- (1+log(x))/x
- Graphing y =:
-
(1+log(x))/x
-
Identical expressions
- (one +log(x))/x
- (1 plus logarithm of (x)) divide by x
- (one plus logarithm of (x)) divide by x
- 1+logx/x
- (1+log(x)) divide by x
-
Similar expressions
- (-1+log(x))/(x-e)
- (-1+log(x))/(x-exp(x))
- (1-log(x))/x
- (-1+log(x))/x
Limit of the function (1+log(x))/x
The solution
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/1 + log(x)\ lim |----------| x->-oo\ x /
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} + 1}{x}\right)$$
Limit((1 + log(x))/x, x, -oo)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to -\infty} x = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} + 1}{x}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(\log{\left(x \right)} + 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to -\infty} \frac{1}{x}$$
=
$$\lim_{x \to -\infty} \frac{1}{x}$$
=
$$0$$
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)
oo/-oo,
i.e. limit for the numerator is
$$\lim_{x \to -\infty}\left(\log{\left(x \right)} + 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to -\infty} x = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} + 1}{x}\right)$$
=
$$\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} \left(\log{\left(x \right)} + 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to -\infty} \frac{1}{x}$$
=
$$\lim_{x \to -\infty} \frac{1}{x}$$
=
$$0$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} + 1}{x}\right) = 0$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} + 1}{x}\right) = 0$$
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
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)} + 1}{x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)} + 1}{x}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} + 1}{x}\right) = 0$$
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
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)} + 1}{x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)} + 1}{x}\right) = 1$$
More at x→1 from the right
The graph