Determine the value for $$ c $$ so that $$ \lim _{x ightarrow-2} f(x) $$ exists. $$ f(x)=\left\{\begin{array}{cc}x^{2}-7, & ext { for } x-2\end{array} ight. $$

Question

Determine the value for $$ c $$ so that $$ \lim _{x ightarrow-2} f(x) $$ exists. $$ f(x)=\left\{\begin{array}{cc}x^{2}-7, & 	ext { for } x<-2 \ -x^{2}+c, & 	ext { for } x>-2\end{array}ight. $$

UpStudy AI · Accepted Answer

**Step 1: Find the left-hand limit (as $$ x \rightarrow -2^- $$)**

For $$ x < -2 $$, we have 
$$
f(x) = x^2 - 7.
$$
So, the left-hand limit is
$$
\lim_{x \to -2^-} f(x) = (-2)^2 - 7 = 4 - 7 = -3.
$$

**Step 2: Find the right-hand limit (as $$ x \rightarrow -2^+ $$)**

For $$ x > -2 $$, we have 
$$
f(x) = -x^2 + c.
$$
Thus, the right-hand limit is 
$$
\lim_{x \to -2^+} f(x) = -(-2)^2 + c = -4 + c = c - 4.
$$

**Step 3: Set the left-hand limit equal to the right-hand limit**

For the limit $$ \lim_{x \rightarrow -2} f(x) $$ to exist, the left-hand and right-hand limits must be equal:
$$
-3 = c - 4.
$$

**Step 4: Solve for $$ c $$**

Add 4 to both sides:
$$
-3 + 4 = c.
$$
Thus,
$$
c = 1.
$$

**Final Answer:**
$$
c = 1.
$$
$$ c = 1 $$

Determine the value for so that exists.

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