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

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

Options:

Correct Answer: 1, Continuous

Solution:

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

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

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

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.

Limit Evaluation – Understanding how to evaluate the limit of a function as it approaches a specific point, particularly using the limit definition and properties of trigonometric functions.
Continuity – Determining if a function is continuous at a point by checking if the limit at that point equals the function's value at that point.