Question

for the function ( f(x) ) shown below, determine ( lim_{x \to -3} f(x) ). ( f(x) = \begin{cases} 3x + 7 & \text{for } x > -3 \\ 7 - x^2 & \text{for } x le -3 end{cases} )

Explanation:

Step1: Find the right - hand limit (as \(x\to - 3^{+}\))

When \(x\to - 3^{+}\), \(x>-3\), so we use the function \(f(x)=3x + 7\).
We substitute \(x=-3\) into \(3x + 7\): \(3\times(-3)+7=-9 + 7=-2\).

Step2: Find the left - hand limit (as \(x\to - 3^{-}\))

When \(x\to - 3^{-}\), \(x\leq - 3\), so we use the function \(f(x)=7 - x^{2}\).
We substitute \(x = - 3\) into \(7-x^{2}\): \(7-(-3)^{2}=7 - 9=-2\).

Step3: Determine the limit

Since the left - hand limit \(\lim_{x\to - 3^{-}}f(x)=-2\) and the right - hand limit \(\lim_{x\to - 3^{+}}f(x)=-2\), by the definition of the limit of a function at a point, if \(\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=L\), then \(\lim_{x\to a}f(x)=L\). Here \(a = - 3\) and \(L=-2\).

Answer:

\(-2\)