Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (8*sin(x)+sin(5*x))/(-sin(4*x)+sin(7*x))
-
Limit of (-sin(2*x)+sin(3*x))/(-sin(4*x)+sin(5*x))
-
Limit of sin(3)^2/sin(2)^2
-
Limit of (-2*sin(x)+sin(2*x))/(5*x^2*log(cos(x)))
- Graphing y =:
- (-1-log(x))/x
-
Identical expressions
- (- one -log(x))/x
- ( minus 1 minus logarithm of (x)) divide by x
- ( minus one minus logarithm of (x)) divide by x
- -1-logx/x
- (-1-log(x)) divide by x
-
Similar expressions
- (1-log(x))/x
- (-1+log(x))/x
Limit of the function (-1-log(x))/x
The solution
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[src]
/-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, dir='-')
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.
$$\lim_{x \to \infty}\left(\frac{- \log{\left(x \right)} - 1}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \log{\left(x \right)} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{1}{x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{1}{x}\right)$$
=
$$0$$
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)
-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.
$$\lim_{x \to \infty}\left(\frac{- \log{\left(x \right)} - 1}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \log{\left(x \right)} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{1}{x}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{1}{x}\right)$$
=
$$0$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{- \log{\left(x \right)} - 1}{x}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \log{\left(x \right)} - 1}{x}\right) = -\infty$$
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
$$\lim_{x \to -\infty}\left(\frac{- \log{\left(x \right)} - 1}{x}\right) = 0$$
More at x→-oo
The graph