Question

Describe the L-Hospital Rule in brief.

Answer 1

Hint: First we will define the L-Hospital Rule and then we will have to define the indeterminate form and then we will define the formula of the L-Hospital rule. After that we will take an example to show the L-hospital rule.

Complete step by step answer:
So, in Calculus, the most important rule is L-Hospital Rule. This rule uses the derivatives to evaluate the limits which involve the indeterminate forms. Let’s understand what an indeterminate form is.
Suppose we have to calculate a limit of $f\left( x \right)$ at $x \to a$ . Then we will first check whether it is an indeterminate form or not by directly putting the value of $x=a$ in the given function. If we get $\dfrac{f\left( a \right)}{g\left( a \right)}=\dfrac{0}{0},\dfrac{\infty }{\infty },{{1}^{\infty }}$ etc. these all are called indeterminate forms. Now, L-Hospital Rule is applicable in the first two cases that is $\dfrac{0}{0},\dfrac{\infty }{\infty }$ .
Let’s see what the L-hospital formula is. Now L-Hospital rule states that if:
$\displaystyle \lim_{x \to c}f\left( x \right)=\displaystyle \lim_{x \to c}g\left( x \right)=0\text{ or }\pm \infty \text{ , g }\!\!'\!\!\text{ }\left( x \right)\ne 0\text{ for all }x\text{ with }x\ne c$ and $\displaystyle \lim_{x \to c}\dfrac{f'\left( x \right)}{g'\left( x \right)}$ exists, then:
$\displaystyle \lim_{x \to c}\dfrac{f\left( x \right)}{g\left( x \right)}=\displaystyle \lim_{x \to c}\dfrac{f'\left( x \right)}{g'\left( x \right)}$
Let’s take an example to see how we apply L-Hospital rule, let’s say we have to solve: $\displaystyle \lim_{x \to 0}\dfrac{\sin x}{x}$ :
So, according to L-hospital rule we will first check if the given function is in the indeterminate form so we will put $x=0$ in the given function: $\Rightarrow \dfrac{\sin x}{x}=\dfrac{\sin 0}{0}=\dfrac{0}{0}$ , now since it is indeterminate form we will apply the L-hospital rule. So, we will just differentiate numerator and denominator separately: $\displaystyle \lim_{x \to 0}\dfrac{\sin x}{x}=\displaystyle \lim_{x \to 0}\dfrac{\sin '\left( x \right)}{\left( x \right)'}=\displaystyle \lim_{x \to 0}\dfrac{\cos x}{1}=\displaystyle \lim_{x \to 0}\cos x$ , now we will find the limit: $\displaystyle \lim_{x \to 0}\cos x=\cos \left( 0 \right)=1$
Therefore, by applying the L-hospital rule we get the limit as 1.

Note: While doing these types of questions that is when we are given a topic to describe, always try and give examples. Remember that the limit of the quotient of a function is equivalent to the limit of the quotient of their derivatives, given that the provided conditions are satisfied.