Question

Determine whether f(x) = 1/(2x+1) is continuous at x = 1/2. Justify using the continuity test.

Answer 1

The function f(x) = 1/(2x+1) is continuous at x = 1/2 because it satisfies all the conditions of continuity: f(x) is defined at x = 1/2, the limit as x approaches 1/2 exists, and this limit is equal to the function's value at x = 1/2.

Step-by-step explanation:

To determine whether f(x) = 1/(2x+1) is continuous at x = 1/2, we must use the continuity test which checks if the following three conditions are met:

1. The function f(x) is defined at x = 1/2.
2. The limit of f(x) as x approaches 1/2 exists.
3. The limit of f(x) as x approaches 1/2 equals f(1/2).

Firstly, we can see that f(x) is defined at x = 1/2 because the denominator 2x+1 will become 2(1/2)+1, which is not equal to 0.

Next, we calculate the limit of f(x) as x approaches 1/2:

\[
\lim_{{x \to 1/2}} f(x) = \lim_{{x \to 1/2}} \frac{1}{2x+1} = \frac{1}{2(1/2)+1} = \frac{1}{2}\]

Last, we determine f(1/2):

f(1/2) = \frac{1}{2(1/2)+1} = \frac{1}{2}

Since all three conditions for continuity are met, f(x) is continuous at x = 1/2. The limit as x approaches 1/2 equals the function's value at x = 1/2, which is 1/2.