Question

a function ( f ) satisfies ( lim_{x \to 3} f(x) = 6 ). which of the following could be the graph of ( f )?

Explanation:

To determine which graph could represent \( f \) with \( \lim_{x \to 3} f(x) = 6 \), we analyze the limit definition: the limit as \( x \) approaches 3 exists and equals 6 if the left - hand limit (LHL) and right - hand limit (RHL) as \( x \to 3 \) are both equal to 6. The value of the function at \( x = 3 \) (i.e., \( f(3) \)) does not affect the limit.

Analyze Option A

• As \( x \) approaches 3 from the left, the \( y \) - value approaches 4 (the open circle at \( x = 3 \) on the left - hand curve is at \( y = 4 \)).
• As \( x \) approaches 3 from the right, the \( y \) - value approaches 6 (the solid line on the right - hand side of \( x = 3 \) starts from a point with \( y = 6 \)).
• Since the left - hand limit (\( \lim_{x\to3^{-}}f(x)=4 \)) and the right - hand limit (\( \lim_{x\to3^{+}}f(x) = 6 \)) are not equal, \( \lim_{x\to3}f(x) \) does not exist. So, Option A is incorrect.

Analyze Option B

• As \( x \) approaches 3 from the left, we look at the curve on the left of \( x = 3 \). The \( y \) - value of the curve approaches 4 (the open circle at \( x = 3 \) on the left - hand curve is at \( y = 4 \))? Wait, no, let's re - examine. Wait, the curve in Option B: as \( x \) approaches 3 from the left, the curve is approaching a value, and as \( x \) approaches 3 from the right, the line is approaching a value. Wait, actually, when we look at the graph in Option B, as \( x \) approaches 3 from the left (along the curve) and from the right (along the line), both the left - hand limit and the right - hand limit are approaching 6? Wait, no, let's correct. Wait, in Option B, when \( x \) approaches 3 from the left, the curve is approaching a value, and when \( x \) approaches 3 from the right, the line is approaching a value. Wait, actually, the key is that for the limit as \( x\to3 \) to be 6, the left - hand limit and right - hand limit as \( x\to3 \) must be 6. In Option B, as \( x \) gets closer to 3 from both the left and the right, the \( y \) - values of the graph (the curve on the left and the line on the right) are approaching 6. The presence of the points (the solid and open dots) at other \( y \) - values (not at \( x = 3 \)) does not affect the limit, because the limit depends on the behavior of the function near \( x = 3 \), not at \( x = 3 \) or at other \( x \) - values.

Analyze Option C (assuming the partial graph)

• From the partial graph, as \( x \) approaches 3 from the left, the \( y \) - value is 6 (the horizontal line), but as \( x \) approaches 3 from the right, the \( y \) - value is approaching a value less than 6 (the line going downwards). So, the left - hand limit \( \lim_{x\to3^{-}}f(x)=6 \) and the right - hand limit \( \lim_{x\to3^{+}}f(x)

eq6 \), so \( \lim_{x\to3}f(x) \) does not exist.

So, the graph in Option B has the property that as \( x \) approaches 3 from both the left and the right, the function values approach 6, so \( \lim_{x\to3}f(x)=6 \).

To determine the correct graph for \( \lim_{x\to3}f(x) = 6 \), we check the left - hand limit (LHL) and right - hand limit (RHL) as \( x\to3 \). The limit exists and equals 6 if \( \text{LHL}=\text{RHL} = 6 \).

• Option A: LHL (as \( x\to3^{-} \)) is 4, RHL (as \( x\to3^{+} \)) is 6. Since \( \text{LHL}

eq\text{RHL} \), the limit does not equal 6.

• Option B: As \( x\to3 \) from both left and right, the function values approach 6 (the limit depends on behavior near \( x = 3 \), not at \( x = 3 \) or other points). So, \( \lim_{x\to3}f(x)=6 \).
• Option C: LHL (as \( x\to3^{-} \)) is 6, but RHL (as \( x\to3^{+} \)) is not 6. So, the limit does not equal 6.

Answer:

B