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 $x=6$.
\[
\lim _{x \rightarrow 6^{-}} f(x)=
\]

Answer

Expert–verified

Hide Steps

Answer

\boxed{\text{The function } f(x) \text{ has a jump discontinuity at } x=6.}

Steps

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.}