Question

How do I find the limit of a rational function?

Answer 1

Hint : The limit of a rational function can be found easily if the given point falls in the domain of the function and is not given the value of not defined at that point. The limit is then calculated by just substituting the value of the given point into the function itself and that will give us the value of the limit for that particular function at a given point.

Complete step-by-step answer :
The limit of a rational function can be found by putting the given point in the function, if the given point is not on the boundary of the domain or the function is not undefined at the given point the process is simple as given above in case the function shows undefined behaviour then we start finding limit of the function from the left and right side limits and putting them equal to each other
$\mathop {\lim }\limits_{x \to {a^ + }} f(x)$ is the right hand limit and $\mathop {\lim }\limits_{x \to {a^ - }} f(x)$ is the left hand limit. If both the values are equal we can say that the limit exists otherwise the limit does not exist.
There is another way which is called as L’Hospital Rule which is defined as
$\mathop {\lim }\limits_{x \to a} \dfrac{{f\left( x \right)}}{{g\left( x \right)}} = \mathop {\lim }\limits_{x \to a} \dfrac{{f'\left( x \right)}}{{g'\left( x \right)}}$
The above formula is only applied when the given both function turn out to be indeterminate or zero and lead to $\left( {\dfrac{0}{0}} \right)$ form or the $\left( {\dfrac{\infty }{\infty }} \right)$ form.

Note : If a limit does not exist at a given point we then try to find the limit of the function using the L’Hospital Rule. In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule provides a technique to evaluate limits of indeterminate forms