Questions: Find the indicated one-sided limits, if they exist. (If an answer does not exist, enter DNE.) f(x) = -x+3 if x ≤ 0 4x+2 if x > 0 lim x → 0+ f(x) = 9 lim x → 0- f(x) = 8

Transcript text: Find the indicated one-sided limits, if they exist. (If an answer does not exist, enter DNE.) \[ \begin{array}{l} f(x)=\left\{\begin{array}{ll} -x+3 & \text { if } x \leq 0 \\ 4 x+2 & \text { if } x>0 \end{array}\right. \\ \lim _{x \rightarrow 0^{+}} f(x)=9 \\ \lim _{x \rightarrow 0^{-}} f(x)=8 \end{array} \]

Solution

Solution Steps

Step 1: Define the Piecewise Function

The function \( f(x) \) is defined as: \[ f(x) = \begin{cases} -x + 3 & \text{if } x \leq 0 \\ 4x + 2 & \text{if } x > 0 \end{cases} \]

Step 2: Calculate the Right-Hand Limit

To find \(\lim_{{x \to 0^+}} f(x)\), we use the definition of \( f(x) \) for \( x > 0 \): \[ f(x) = 4x + 2 \] Substituting \( x = 0 \) from the right: \[ \lim_{{x \to 0^+}} f(x) = 4(0) + 2 = 2 \]

Step 3: Calculate the Left-Hand Limit

To find \(\lim_{{x \to 0^-}} f(x)\), we use the definition of \( f(x) \) for \( x \leq 0 \): \[ f(x) = -x + 3 \] Substituting \( x = 0 \) from the left: \[ \lim_{{x \to 0^-}} f(x) = -0 + 3 = 3 \]