Question

Is the function continuous at x = -17

Answer 1

For a function to be continuous at an x-value, say -17, you need to make sure two things line up:

The limit from the left equals the limit from the right.

     $\lim_{x \to -17^{-}} f(x) = \lim_{x \to -17^{+}} f(x)$

This limit equals the functions value.

    $\lim_{x \to -17} f(x) = f(-17)$

The left hand limit involves the first piece, f(x) = 20x + 1:

    $\begin{aligned} \lim_{x \to -17^{-}} f(x) &= \lim_{x \to -17^{-}} (20x+1)\\[0.5em]&= 20(-17)+1\\[0.5em]&= -339\endaligned}$

The right hand limit invovles the second piece, f(x) = -10x^2:

    $\begin{aligned} \lim_{x \to -17^{+}} f(x) &= \lim_{x \to -17^{+}} (-10x^2)\\[0.5em]&= -10\cdot (-17)^2\\[0.5em]&= -2890\endaligned}$

Since the two one-sided limits don't match, the function is not continuous at x=-17.