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

Calculus

1. Limits and Continuity

Continuity

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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

To determine if the function f(x) is continuous at x = -2, we need to check three conditions: the function must be defined at x = -2, the limit of the function as x approaches -2 must exist, and the limit must equal the function value at x = -2.

First, check if f(x) is defined at x = -2. Look at the graph and see if there is a point at x = -2. In the graph, there is a solid dot at x = -2, indicating that f(-2) is defined.

Next, determine the limit of f(x) as x approaches -2 from both sides. Observe the graph to see if the function approaches the same value from the left and right as x approaches -2. The graph shows that the function approaches the same value from both sides.

Compare the limit of f(x) as x approaches -2 with the value of f(-2). Since the limit from both sides equals the value of the function at x = -2, the function is continuous at this point.

Conclude that f(x) is continuous at x = -2 because all three conditions for continuity are satisfied: the function is defined at x = -2, the limit exists, and the limit equals the function value.