Evaluate the limit lim (x -> 0) (sin(x)/x). Is the function continuous at x = 0?

1 question

1 item

Q: Evaluate the limit lim (x -> 0) (sin(x)/x). Is the function continuous at x = 0?

Solution: The limit is 1, and if we define f(0) = 1, then f(x) is continuous at x = 0.

Steps: 6

Step 1: Understand the limit we want to evaluate: lim (x -> 0) (sin(x)/x). This means we want to see what happens to the value of sin(x)/x as x gets very close to 0.
Step 2: Recognize that directly substituting x = 0 into sin(x)/x gives us 0/0, which is undefined. So we need to find another way to evaluate the limit.
Step 3: Use a known limit result: lim (x -> 0) (sin(x)/x) = 1. This is a standard limit in calculus.
Step 4: Since the limit as x approaches 0 is 1, we can say that lim (x -> 0) (sin(x)/x) = 1.
Step 5: To check if the function is continuous at x = 0, we need to define f(0). If we set f(0) = 1, then the function is continuous at x = 0 because the limit equals the function value at that point.
Step 6: Conclude that the limit is 1 and if we define f(0) = 1, then f(x) is continuous at x = 0.