Animated Solution for Mathematics - Limits, Continuity and Differentiability: Let $f:R\to [0,\infty )$ be such that ${\lim }_{x\to 5}f(x)$ exists and ${\lim }_{x\to 5}\frac{(f(x){)}^2-9}{\sqrt{|x-5|}}=0$. Then ${\lim }_{x\to 5}f(x)$ equals

We are given a function $f:\mathbb{R}\to [0,\infty )$ and the limit equation:

Our objective is to determine the value of ${\lim }_{x\to 5}f(x)$. We begin by observing the behavior of the expression as $x$ approaches $5$.

Let the denominator be defined as $D(x)=\sqrt{|x-5|}$. As $x\to 5$, the term $|x-5|$ approaches $0$, which implies that $D(x)\to 0$.

If the denominator of a fraction approaches zero, the only way the entire expression can converge to a finite value (in this case, $0$) is if the numerator also approaches zero. If the numerator were a non-zero constant, the limit would diverge to infinity.

Since the limit of the entire expression is $0$, we must satisfy the condition:

By applying the properties of limits, we distribute the limit across the subtraction:

Since the limit of the constant $9$ is $9$, we rearrange the equation to find:

Taking the square root of both sides, we find that ${\lim }_{x\to 5}f(x)$ must be either $3$ or $-3$.

We must now apply the codomain constraint provided in the problem statement: $f:\mathbb{R}\to [0,\infty )$. This constraint dictates that for all $x$ in the domain, $f(x)\ge 0$.

Because the function is strictly non-negative, its limit as $x$ approaches $5$ cannot be negative. Consequently, we reject $-3$ as a valid solution.

The only remaining possibility is that the limit is $3$. Therefore, the final answer is: