Answer
Using theorem, we know that limit does not exist for the quotient of two functions if the function in the denominator approaches to $0$.
But according to the question $\lim_\limits{x\to a }\dfrac{f(x)}{g(x)}$ exists.
So, there must be a common factor in both the functions which approaches to $0$ as $x\to a$. Since, this common factor from the numerator cancels the factor in the denominator, we get a finite value for the limit.
Since, this factor is in $f(x)$ and this factor approaches to $0$ as $x\to a$.
If $x\to a$, $f(x)$ also approaches to $0$.
Hence, $\lim_\limits{x\to a}f(x)=0$
Work Step by Step
Using theorem, we know that limit does not exist for the quotient of two functions if the function in the denominator approaches to $0$.
But according to the question $\lim_\limits{x\to a }\dfrac{f(x)}{g(x)}$ exists.
So, there must be a common factor in both the functions which approaches to $0$ as $x\to a$. Since, this common factor from the numerator cancels the factor in the denominator, we get a finite value for the limit.
Since, this factor is in $f(x)$ and this factor approaches to $0$ as $x\to a$.
If $x\to a$, $f(x)$ also approaches to $0$.
Hence, $\lim_\limits{x\to a}f(x)=0$
