Question

How do you find the limit $\dfrac{{\sqrt {49 + h} - 7}}{h}$ as h approaches $0$ using L’Hospital’s rule?

Answer 1

Hint: The given question requires us to evaluate a limit using the L’Hospital’s rule. There are various methods and steps to evaluate a limit. Some of the common steps while solving limits involve rationalization and applying some basic results on frequently used limits. L’Hospital’s rule involves solving limits of indeterminate form by differentiating both numerator and denominator with respect to the variable separately and then applying the required limit.

Complete step by step answer:
We have to evaluate the limit $\lim \dfrac{{\sqrt {49 + h} - 7}}{h}$ as $h \to 0$ using L’Hospital’s rule. So, if we put the limit x tending to zero into the expression $\dfrac{{\sqrt {49 + h} - 7}}{h}$, we get an indeterminate form limit. Hence, L’Hospital’s rule can be applied here to find the value of the concerned limit.
So, Applying L’Hospital’s rule, we have to differentiate both numerator and denominator with respect to the variable x separately and then apply the limit.
Hence, $\mathop {\lim }\limits_{h \to 0} \dfrac{{\sqrt {49 + h} - 7}}{h}$
Now, the derivative of $\sqrt {49 + h} $ can be evaluated as $\dfrac{1}{{2\sqrt {49 + h} }}$ using the chain rule of differentiation and the derivative of h with respect to h is $1$ using the power rule of differentiation. So, applying the L’Hospital’s rule to find the given limit, we get,
$ = \mathop {\lim }\limits_{h \to 0} \dfrac{1}{{2\sqrt {49 + h} }}$
Substituting the limit, we get,
$ = \dfrac{1}{{2\sqrt {49 + 0} }}$
Simplifying the calculations, we get,
$ = \dfrac{1}{{2\left( 7 \right)}}$
$ = \dfrac{1}{{14}}$
So, the value of the limit $\dfrac{{\sqrt {49 + h} - 7}}{h}$ as $h \to 0$ is $\dfrac{1}{{14}}$ .

Note: The given question can also be solved in a different method. We must remember some basic limits to solve such questions involving complex limits. We can also apply some basic algebraic rules like multiplying the term with its conjugate so as to remove the square root entity and simplify the expression.