Find the limit using the graph of f ( x ) f\left(x\right) f ( x ) shown. lim x → 4 f ( x ) \lim_{x\rarr4}f\left(x\right) lim x → 4 f ( x )

In this problem, we're asked to find the limit using the graph of f of x shown below. Now here we're specifically asked to find the limit of f of x as x approaches 4. So looking at our graph here, we want to look at what our function is doing as x gets really, really close to a value of 4. So looking at my graph here coming in from that left side, as x gets really, really close to 4, I see that my function seems to be approaching a value of negative 2. Then coming in from the other side, I see the same thing. As x gets really, really close to 4, my function seems to be approaching a value of negative 2. So here the limit of f of x as x approaches 4 is equal to negative 2. Now a mistake that you might have made here is thinking that we don't even have a limit here because there's a hole in our graph. Now it's easy to get tripped up by that the first time you see it. But remember that it doesn't matter what the value of our function is at 4, but just what it's doing as x approaches 4 as it gets really, really close to it but isn't actually equal to it. So it doesn't matter that our graph isn't actually defined for our x value of 4. It just matters that as x gets really close to 4, our function is approaching a value of negative 2, and that's all there is to it. Let me know if you have any questions there, and I'll see you in the next one.

1. Limits and Continuity

Introduction to Limits

Business Calculus

1. Limits and Continuity

Introduction to Limits

Struggling with Business Calculus?

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Multiple Choice

Find the limit using the graph of $f\left(x\right)$ shown.
$\lim_{x\rarr4}f\left(x\right)$

$2$

$-2$

$0$

Unable to determine

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the limit of f(x) as x approaches 4 using the graph provided. This involves analyzing the behavior of the function f(x) as x gets closer to 4 from both the left and the right sides.

Step 2: Observe the graph near x = 4. Look at the values of f(x) as x approaches 4 from the left (x < 4) and from the right (x > 4). The graph shows a continuous curve leading to a specific y-value near x = 4.

Step 3: Identify the y-value that the graph approaches as x approaches 4. From the graph, trace the curve to see where it converges near x = 4. Note that the open circle at x = 4 indicates the function value at x = 4 is not defined, but the limit depends on the behavior of the graph as x approaches 4.

Step 4: Confirm that the left-hand limit and right-hand limit are equal. Check the graph to ensure that the y-value approached by the function is the same whether x approaches 4 from the left or the right. This confirms the existence of the limit.

Step 5: Conclude the limit. Based on the graph, the y-value that f(x) approaches as x approaches 4 is −2. This is the limit of f(x) as x approaches 4.