Use the graph of f ( x ) f\left(x\right) f ( x ) to determine if the function is continuous or discontinuous at x = c x=c x = c . c = − 2 c=-2 c = − 2

Here we wanna use the graph of f of x to determine if our function is continuous or discontinuous at x equals c. Now here we're looking at x equals negative 2 because that's our specified value of c. Now looking at my graph here, I know that if I trace along my function at x equals 2, I don't have to lift my pen at all. So I know that this function is continuous. Now, of course, we can also verify this by looking at both the limit and the function value at this point and we would see that they are equal to each other because they are both 2. Thanks for watching, and let me know if you have questions.

1. Limits and Continuity

Continuity

Business Calculus

1. Limits and Continuity

Continuity

Struggling with Business Calculus?

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Multiple Choice

Use the graph of $f\left(x\right)$ to determine if the function is continuous or discontinuous at $x=c$ .
$c=-2$

Continuous

Discontinuous

Verified step by step guidance

Step 1: Recall the definition of continuity at a point x = c. A function f(x) is continuous at x = c if the following three conditions are satisfied: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c).

Step 2: Analyze the graph at x = -2. Check if there is a point on the graph at x = -2. From the graph, observe that there is a filled circle at x = -2, indicating that f(-2) is defined.

Step 3: Determine if the limit of f(x) as x approaches -2 exists. Observe the behavior of the graph as x approaches -2 from both the left and the right. The graph approaches the same y-value from both sides, indicating that the limit exists.

Step 4: Verify if the limit of f(x) as x approaches -2 is equal to f(-2). From the graph, the y-value of the function at x = -2 matches the value approached by the graph from both sides, satisfying the third condition for continuity.

Step 5: Conclude that the function f(x) is continuous at x = -2 because all three conditions for continuity are satisfied.