Determine whether the function f(x) is continuous at x=3. f(x)={[7-2x^(2)",",x 3]:} f(x) is continuous at x=3 f(x) is discontinuous at x=3

Question

Determine whether the function  f(x) is continuous at  x=3.  f(x)={[7-2x^(2)",",x <= 3],[-8-x",",x > 3]:}  f(x) is continuous at  x=3  f(x) is discontinuous at  x=3

Bytelearn · Accepted Answer

Check Function Definition: To determine if the function f(x) is continuous at x=3, we need to check three conditions: 1. The function is defined at x=3. 2. The limit of f(x) as x approaches 3 exists. 3. The limit of f(x) as x approaches 3 is equal to the function value at x=3.Find Left Limit: First, let's check if the function is defined at x=3. We look at the piece of the function that applies when x is less than or equal to 3, which is f(x) = 7 - 2x^2. Plugging in x=3, we get f(3) = 7 - 2(3)^2 = 7 - 18 = -11. So, the function is defined at x=3.Find Right Limit: Next, we need to find the limit of f(x) as x approaches 3 from the left. This means we will use the piece of the function that applies when x is less than or equal to 3, which is f(x) = 7 - 2x^2. The limit as x approaches 3 from the left is the same as the function value at x=3, which we already calculated as -11.Compare Limits: Now, we need to find the limit of f(x) as x approaches 3 from the right. This means we will use the piece of the function that applies when x is greater than 3, which is f(x) = -8 - x. Plugging in x=3, we get the limit as x approaches 3 from the right to be -8 - 3 = -11.Check Continuity: Since the limit from the left and the limit from the right both exist and are equal to each other, the limit of f(x) as x approaches 3 exists and is equal to -11.Finally, we compare the limit of f(x) as x approaches 3, which is -11, to the function value at x=3, which is also -11. Since these two values are equal, the function is continuous at x=3.

Determine whether the function $f(x)$ is continuous at $x=3$ . $\newline$ $f(x)=\left\{\begin{array}{ll} 7-2 x^{2}, & x \leq 3 \\ -8-x, & x>3 \end{array}\right.$ $\newline$ $f(x)$ is continuous at $x=3$ $\newline$ $f(x)$ is discontinuous at $x=3$

Full solution

More problems from Find derivatives of logarithmic functions

Related topics