Problem
A function $f(x)$ is said to have a jump discontinuity at $x=a$ if: 1. $\lim _{x \rightarrow a^{-}} f(x)$ exists. 2. $\lim _{x \rightarrow a^{+}} f(x)$ exists. 3. The left and right limits are not equal. Let $f(x)=\left\{\begin{array}{ll}x^{2}+4 x+3 & \text { if } x<6 \\ 17 & \text { if } x=6 \\ -7 x+6 & \text { otherwise }\end{array}\right.$ Show that $f(x)$ has a jump discontinuity at $x=6$ by calculating the limits from the left and right at $x=6$. \[ \lim _{x \rightarrow 6^{-}} f(x)= \]
Solution
Step 1 ：Find the left limit: \(\lim_{x \rightarrow 6^{-}} f(x) = 6^2 + 4(6) + 3 = 63\)
Step 2 ：Find the right limit: \(\lim_{x \rightarrow 6^{+}} f(x) = -7(6) + 6 = -36\)
Step 3 ：\boxed{\text{The function } f(x) \text{ has a jump discontinuity at } x=6.}
