For what value(s) of $k$ is $f$ continuous on $\mathbb{R}$

Question

For what value(s) of $k$ is $f$ continuous on $\mathbb{R}$ ?

Define $f \colon \mathbb{R} \rightarrow \mathbb{R}$. via \begin{array}{@{} r @{} c @{} l @{} } \\[1ex] &f(x) &{}=\displaystyle \begin{cases} \frac{10x^2-16x-8}{x-2} &\text{if } x \neq 2 \\ k &\text{if } x=2 \end{cases} \end{array}

So i was able to see that the $$\lim_{x\to 2} f(x)=24$$. so does that mean k must be 24?

Answer 1

Yes k must be 24 for the function to be continuous. It is the definition of continuity: A function is continuous if the limit of its value as x approaches a point is equal to the value of the function at that point, or, in notation, $$\lim_{x\rightarrow p}{f(x)}=f(p)$$