Question

a function f(x) is said to have a jump discontinuity at x = a if: 1. lim┬(x→a⁻)⁡〖f(x) exists.〗 2. lim┬(x→a⁺)⁡〖f(x) exists.〗 3. the left and right limits are not equal. let f(x)={█(x^2 + 7x + 1 if x < 8@15 if x = 8@-7x + 4 otherwise)〗 show that f(x) has a jump discontinuity at x = 8 by calculating the limits from the left and right at x = 8. lim┬(x→8⁻)⁡〖f(x)=〗□( ) lim┬(x→8⁺)⁡〖f(x)=〗□( ) now for fun, try to graph f(x).

Explanation:

Step1: Calculate left - hand limit

For $x\to8^{-}$, we use the formula $f(x)=x^{2}+7x + 1$ (since $x\lt8$).
$\lim_{x\to8^{-}}f(x)=\lim_{x\to8^{-}}(x^{2}+7x + 1)$.
Substitute $x = 8$ into $x^{2}+7x + 1$: $8^{2}+7\times8 + 1=64 + 56+1=121$.

Step2: Calculate right - hand limit

For $x\to8^{+}$, we use the formula $f(x)=-7x + 4$ (since $x\gt8$, and for the right - hand limit we consider values slightly greater than 8).
$\lim_{x\to8^{+}}f(x)=\lim_{x\to8^{+}}(-7x + 4)$.
Substitute $x = 8$ into $-7x + 4$: $-7\times8+4=-56 + 4=-52$.

Answer:

$\lim_{x\to8^{-}}f(x)=121$, $\lim_{x\to8^{+}}f(x)=-52$