Multiple choice

Which one of the following functions is continuous at x = 3?

  1. $f(x)= \begin{cases} 2 \text{, if x = 3} \\\\ x -1 \text{ if }\ x \gt 3 \\\\ \dfrac{x+3}{3} \text{ if }\ x \gt 3 \end{cases} $
  2. $f(x)= \begin{cases} 4 \text{, if x = 3} \\\\ 8-x\ \text{if}\ x \ne 3 \end{cases} $
  3. $f(x)= \begin{cases} x+3 \text{, if } x = 3 \\\\ x -4 \text{ if } x \ne 3 \end{cases} $
  4. $f(x) = \dfrac{1}{x^3 - 27} \text{, if } x \ne 3$
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A Correct answer
Explanation

$ \lim_{x \to 3+} f(x) = \lim_{x \to 3+} (x -1) = 2 = f(3)$ $ \lim_{x \to 3-} f(x) = \lim_{x \to 3-} \left(\dfrac{x+3}{3} \right)= 2 = f(3)$ $\therefore$ $f(x)$is continuous at x = 3.

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