Use the given graph to determine the value of the indicated limits. (a) \( \lim _{x \rightarrow-5} f(x) \) (b) \( \lim _{x \rightarrow-4} f(x) \) (a) Select the correct choice below, and if necessary, fill in the answer box to complete your choice. A. lim \( f(x)=\square \) (o) B. The limit does

I'd be happy to help you determine the values of the indicated limits based on the graph of \( f(x) \). However, since I can't see the graph, I'll guide you through the general steps to evaluate these limits. If you can provide specific details about the behavior of \( f(x) \) near \( x = -5 \) and \( x = -4 \), I can offer a more precise answer. ### (a) \( \lim_{x \rightarrow -5} f(x) \) **Steps to Determine the Limit:** 1. **Approach from the Left (\( x \rightarrow -5^- \)):** - Look at the value that \( f(x) \) is approaching as \( x \) gets closer to \(-5\) from values less than \(-5\). 2. **Approach from the Right (\( x \rightarrow -5^+ \)):** - Examine the value that \( f(x) \) is approaching as \( x \) approaches \(-5\) from values greater than \(-5\). 3. **Compare Both Sides:** - **If both one-sided limits exist and are equal**, then \( \lim_{x \rightarrow -5} f(x) \) exists and is equal to that common value. - **If the one-sided limits are different or at least one doesn't exist**, then the limit does not exist. **Possible Scenarios:** - **A.** \( \lim_{x \rightarrow -5} f(x) = L \), where \( L \) is a specific number. - **B.** The limit does not exist. *Select the appropriate option based on the behavior of the graph near \( x = -5 \). If the graph approaches the same value from both sides, choose option A and fill in the value. If not, choose option B.* ### (b) \( \lim_{x \rightarrow -4} f(x) \) **Steps to Determine the Limit:** 1. **Approach from the Left (\( x \rightarrow -4^- \)):** - Observe the behavior of \( f(x) \) as \( x \) approaches \(-4\) from values less than \(-4\). 2. **Approach from the Right (\( x \rightarrow -4^+ \)):** - Observe the behavior of \( f(x) \) as \( x \) approaches \(-4\) from values greater than \(-4\). 3. **Compare Both Sides:** - **If both one-sided limits exist and are equal**, then \( \lim_{x \rightarrow -4} f(x) \) exists and is equal to that value. - **If the one-sided limits differ or one doesn't exist**, then the limit does not exist. **Possible Scenarios:** - **A.** \( \lim_{x \rightarrow -4} f(x) = M \), where \( M \) is a specific number. - **B.** The limit does not exist. *Again, choose the appropriate option based on the graph's behavior near \( x = -4 \).* ### Example Illustration Suppose the graph of \( f(x) \) near \( x = -5 \) looks like this: - As \( x \) approaches \(-5\) from the left, \( f(x) \) approaches \( 3 \). - As \( x \) approaches \(-5\) from the right, \( f(x) \) approaches \( 3 \). In this case: - \( \lim_{x \rightarrow -5} f(x) = 3 \) (Choose Option A and fill in \( 3 \)). Alternatively, if: - As \( x \) approaches \(-4\) from the left, \( f(x) \) approaches \( 2 \). - As \( x \) approaches \(-4\) from the right, \( f(x) \) approaches \( 5 \). Then: - \( \lim_{x \rightarrow -4} f(x) \) does not exist because the left-hand and right-hand limits are different (Choose Option B). ### Final Tips - **Continuity Check:** If \( f(x) \) is continuous at \( x = a \), then \( \lim_{x \rightarrow a} f(x) = f(a) \). - **Discontinuities:** Look for jumps, holes, or vertical asymptotes, as these often affect the existence of limits. If you can describe the graph's behavior around \( x = -5 \) and \( x = -4 \), I can provide more specific guidance!