Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-2+x)/(-2+x^2-x)
-
Limit of cos(5*x)*sin(2*x)/tan(x)
-
Limit of 3*sin(x)^2/(4*x)
-
Limit of log(1+e^x)
-
Identical expressions
- log(x)/(x*log(ten))
- logarithm of (x) divide by (x multiply by logarithm of (10))
- logarithm of (x) divide by (x multiply by logarithm of (ten))
- log(x)/(xlog(10))
- logx/xlog10
- log(x) divide by (x*log(10))
Limit of the function log(x)/(x*log(10))
The solution
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/ log(x) \ lim |---------| x->oo\x*log(10)/
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right)$$
Limit(log(x)/((x*log(10))), x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} \log{\left(x \right)} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(x \log{\left(10 \right)}\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} x \log{\left(10 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{x \log{\left(10 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{x \log{\left(10 \right)}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty} \log{\left(x \right)} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(x \log{\left(10 \right)}\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \log{\left(x \right)}}{\frac{d}{d x} x \log{\left(10 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{x \log{\left(10 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{1}{x \log{\left(10 \right)}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)}}{x \log{\left(10 \right)}}\right) = 0$$
More at x→-oo
The graph