Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-5+7*n)/(2+n)
-
Limit of (-64+4^x)/(-3+x)
-
Limit of (-7+3*x^2+5*x)/(1+x+3*x^2)
-
Limit of (-2+2*x^3+7*x)/(-4-x+3*x^3)
- Graphing y =:
- x+log(x)/x
-
Identical expressions
- x+log(x)/x
- x plus logarithm of (x) divide by x
- x+logx/x
- x+log(x) divide by x
-
Similar expressions
- (e^x+log(x))/x^100
- x+log(x)/(x*log(10))
- (x+log(x))/(x-log(x))
- (x+log(x))/x
- x-log(x)/x
Limit of the function x+log(x)/x
The solution
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/ log(x)\ lim |x + ------| x->oo\ x /
$$\lim_{x \to \infty}\left(x + \frac{\log{\left(x \right)}}{x}\right)$$
Limit(x + log(x)/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x^{2} + \log{\left(x \right)}\right) = \infty$$
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.
$$\lim_{x \to \infty}\left(x + \frac{\log{\left(x \right)}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{x^{2} + \log{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{2} + \log{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(2 x + \frac{1}{x}\right)$$
=
$$\lim_{x \to \infty}\left(2 x + \frac{1}{x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty}\left(x^{2} + \log{\left(x \right)}\right) = \infty$$
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.
$$\lim_{x \to \infty}\left(x + \frac{\log{\left(x \right)}}{x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to \infty}\left(\frac{x^{2} + \log{\left(x \right)}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x^{2} + \log{\left(x \right)}\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(2 x + \frac{1}{x}\right)$$
=
$$\lim_{x \to \infty}\left(2 x + \frac{1}{x}\right)$$
=
$$\infty$$
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
$$\lim_{x \to \infty}\left(x + \frac{\log{\left(x \right)}}{x}\right) = \infty$$
$$\lim_{x \to 0^-}\left(x + \frac{\log{\left(x \right)}}{x}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x + \frac{\log{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x + \frac{\log{\left(x \right)}}{x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x + \frac{\log{\left(x \right)}}{x}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x + \frac{\log{\left(x \right)}}{x}\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(x + \frac{\log{\left(x \right)}}{x}\right) = \infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(x + \frac{\log{\left(x \right)}}{x}\right) = -\infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(x + \frac{\log{\left(x \right)}}{x}\right) = 1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(x + \frac{\log{\left(x \right)}}{x}\right) = 1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(x + \frac{\log{\left(x \right)}}{x}\right) = -\infty$$
More at x→-oo
The graph