Question

How do you find the limit of $\dfrac{{{x}^{2}}}{\ln x}$ as $x$ approaches infinity?

Answer 1

In this question we have been asked to find the limit of the given function $\dfrac{{{x}^{2}}}{\ln x}$ as $x$ approaches infinity. Here we are going to use the L'Hospital rule which states that when the numerator and denominator of the function tend to an undefined form then we can find the limit by differentiating the numerator and denominator individually.

Complete step by step solution:
Now considering from the question we have been asked to find the limit of the given function $\dfrac{{{x}^{2}}}{\ln x}$ as $x$ approaches infinity.
Here we are going to use the L'Hospital rule which states that when the numerator and denominator of the function tend to an undefined form then we can find the limit by differentiating the numerator and denominator individually. Many of us have learnt about this while learning basic concepts of limits.
Now we will first evaluate the value of the numerator and denominator of the given function $\dfrac{{{x}^{2}}}{\ln x}$ when $x$ approaches infinity. Its value will be in the form of $\dfrac{\infty }{\infty }$ .
Hence we will differentiate the numerator and denominator of the function individually. After doing that we will have
$\Rightarrow \dfrac{\dfrac{d}{dx}{{x}^{2}}}{\dfrac{d}{dx}\left( \ln x \right)}=\dfrac{2x}{\left( \dfrac{1}{x} \right)}\Rightarrow 2{{x}^{2}}$
Now we will evaluate the limit of this expression. We will have $\underset{x\to \infty }{\mathop{\lim }}\,2{{x}^{2}}=\infty$ .
Therefore we can conclude that the limit of the given function $\dfrac{{{x}^{2}}}{\ln x}$ as $x$ approaches infinity is $\infty$.

Note: During answering questions of this type we should be sure with our concepts that we are applying and the calculations that we are performing in the process. If we are aware of all the basic concepts then any question related to the topic becomes easy and you can solve them, so I suggest you learn the concept well. Similarly we can apply the L’Hospital to any function in the form of $\dfrac{\infty }{\infty }$ or $\dfrac{0}{0}$ .