Question

for the function ( f ) whose graph is given, state the value of each quantity, if it exists. (if an answer does not exist, enter dne.)
(a) ( limlimits_{x \to dots} f(x) ) (with a text box labeled \enter a number\ and an x indicating an error)
(b) ( limlimits_{x \to dots} f(x) )
(graph: coordinate plane with ( x )-axis marked 0, 2, 4 and ( y )-axis marked 0, 2, 4, showing a curve with a minimum near ( x=2 ), a point near ( x=4 ), etc.)

Explanation:

To solve the limit problems for the function \( f(x) \) from its graph, we analyze the behavior of the function as \( x \) approaches the given value (assuming part (a) is \( \lim_{x \to 2} f(x) \) and part (b) is \( \lim_{x \to 4} f(x) \) based on the graph):

Part (a): \( \lim_{x \to 2} f(x) \)

Step 1: Analyze left-hand limit (\( x \to 2^- \))

As \( x \) approaches \( 2 \) from the left, observe the \( y \)-value the graph approaches. From the left, the function approaches \( 1 \) (or the lowest point near \( x=2 \)).

Step 2: Analyze right-hand limit (\( x \to 2^+ \))

As \( x \) approaches \( 2 \) from the right, the graph also approaches \( 1 \) (since the curve near \( x=2 \) has a minimum at \( y \approx 1 \)).

Step 3: Compare left and right limits

Since \( \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = 1 \), the two-sided limit exists and equals \( 1 \).

Part (b): \( \lim_{x \to 4} f(x) \)

Step 1: Analyze left-hand limit (\( x \to 4^- \))

As \( x \) approaches \( 4 \) from the left, the graph approaches \( 4 \) (the \( y \)-value of the open circle at \( x=4 \)).

Step 2: Analyze right-hand limit (\( x \to 4^+ \))

As \( x \) approaches \( 4 \) from the right, the graph also approaches \( 4 \) (the curve continues from \( x=4 \) with \( y \)-value \( 4 \)).

Step 3: Compare left and right limits

Since \( \lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = 4 \), the two-sided limit exists and equals \( 4 \).

Final Answers

(a) \( \boldsymbol{1} \)
(b) \( \boldsymbol{4} \)

Answer:

To solve the limit problems for the function \( f(x) \) from its graph, we analyze the behavior of the function as \( x \) approaches the given value (assuming part (a) is \( \lim_{x \to 2} f(x) \) and part (b) is \( \lim_{x \to 4} f(x) \) based on the graph):

Part (a): \( \lim_{x \to 2} f(x) \)

Step 1: Analyze left-hand limit (\( x \to 2^- \))

As \( x \) approaches \( 2 \) from the left, observe the \( y \)-value the graph approaches. From the left, the function approaches \( 1 \) (or the lowest point near \( x=2 \)).

Step 2: Analyze right-hand limit (\( x \to 2^+ \))

As \( x \) approaches \( 2 \) from the right, the graph also approaches \( 1 \) (since the curve near \( x=2 \) has a minimum at \( y \approx 1 \)).

Step 3: Compare left and right limits

Since \( \lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = 1 \), the two-sided limit exists and equals \( 1 \).

Part (b): \( \lim_{x \to 4} f(x) \)

Step 1: Analyze left-hand limit (\( x \to 4^- \))

As \( x \) approaches \( 4 \) from the left, the graph approaches \( 4 \) (the \( y \)-value of the open circle at \( x=4 \)).

Step 2: Analyze right-hand limit (\( x \to 4^+ \))

As \( x \) approaches \( 4 \) from the right, the graph also approaches \( 4 \) (the curve continues from \( x=4 \) with \( y \)-value \( 4 \)).

Step 3: Compare left and right limits

Since \( \lim_{x \to 4^-} f(x) = \lim_{x \to 4^+} f(x) = 4 \), the two-sided limit exists and equals \( 4 \).

Final Answers

(a) \( \boldsymbol{1} \)
(b) \( \boldsymbol{4} \)