Typical Calculus Exam Question: Continuity

This Problem of the Week is a typical exam question about continuity, in particular about a removable discontinuity. You should expect a similar one on your exam(s), since it combines what we’ve been practicing to compute limits with the idea of continuity.

Detailed Solution

The key realization for this type of problem is that for a the function f to be continuous at $x=a,$ we must have \[f(a) = \lim_{x \to a}f(x)\] In this case, that means we must have \begin{align*} f(2) &= \lim_{x \to 2}f(x) \\[8px] &= \lim_{x \to 2}\frac{x^2 + 3x-10}{x-2} \end{align*}

We found that very limit in last week’s Problem of the Week. For completeness, here’s the calculation again, making use of the fact that we can factor the numerator: \begin{align*} C = f(2) &= \lim_{x \to 2}\dfrac{x^2 + 3x-10}{x-2} \\[8px] &= \lim_{x \to 2}\frac{(x-2)(x+5)}{x-2} \\[8px] &= \lim_{x \to 2}\frac{\cancel{(x-2)}(x+5)}{\cancel{(x-2)}} \\[8px] &= \lim_{x \to 2}(x+5) = 7 \implies \text{ (D)} \quad \cmark \end{align*}

As we saw in last week’s problem, $y = \dfrac{x^2 + 3x-10}{x-2}$ is a line with a hole at $(2, 7).$ We’re simply filling in this hole to make the function continuous at $x=2,$ which is what makes this a “removable discontinuity.” As you can see, if you’ve gotten good at computing limits (through practice!), then this problem is entirely straightforward.

We have much more discussion about the crucial concept of Continuity, and other types of typical Continuity problems, in our Continuity and Intermediate Value Theorem section. In particular, there are more practice problems similar to this one (super-common on exams!) on the screen Discontinuity Types; Removable Discontinuities. It’s all free, just to support your learning this material well so you can apply it in whatever ways you’d like.

For now, what continuity, or other Calculus, questions are you working on and could use some help with? Please let us know below, or (better) over on our Forum where it’s much easier to write math.

We hope to see you there!