Question

How do you find the limit of $\mathop {\lim }\limits_{x \to \infty } \dfrac{x}{{\ln x}}$ ?

Answer 1

In this question we have to find the limit of the given function, we can evaluate the limit by using L’hospital rule, which states that the if the function tend to an undefined form we will get the value of limit by differentiating the numerator and denominator individually.

Complete step by step solution:
A limit is defined as a function that has some value that approaches the input. A limit of a function is represented as :
$\mathop {\lim }\limits_{x \to n} f\left( x \right) = L$,
Here lim refers to limit, it generally describes that the real valued function$f\left( x \right)$tends to attain the limit L as $x$ tends to n and is denoted by an arrow.
Now given function is $\mathop {\lim }\limits_{x \to \infty } \dfrac{x}{{\ln x}}$,
If we directly apply the limits then the result will be an indeterminate form $\dfrac{\infty }{\infty }$,
So, we will apply L’hospital rule which states that if the function tends to an undefined form we will get the value of limit by differentiating the numerator and denominator individually.
Now differentiating both numerator and denominator we get,
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{x}{{\ln x}} = \mathop {\lim }\limits_{x \to \infty } \dfrac{{\dfrac{d}{{dx}}x}}{{\dfrac{d}{{dx}}\ln x}}$,
Now applying derivatives we get,
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{x}{{\ln x}} = \mathop {\lim }\limits_{x \to \infty } \dfrac{1}{{\dfrac{1}{x}}}$,
Now simplifying we get,
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{x}{{\ln x}} = \mathop {\lim }\limits_{x \to \infty } x$,
Now applying limits directly we get,
$\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{x}{{\ln x}} = \infty$,
So, the value of the limit is $\infty$.

$\therefore$ The value of limit for the given function $\mathop {\lim }\limits_{x \to \infty } \dfrac{x}{{\ln x}}$will be equal to $\infty$.

Note:
Remember that L’hospital rule can be applied if the given function is fraction of two functions in the form $\dfrac{{f\left( x \right)}}{{g\left( x \right)}}$, and it is very important to notice when we substitute the value of $x$ the function must evaluate to either $\dfrac{0}{0}$ or $\dfrac{\infty }{\infty }$ as these are two types of indeterminate forms, and we cannot apply the rule if the limit question is not in an indeterminate form.