Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.

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Question: Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.

Options:

Correct Answer: 1, Continuous

Solution:

The limit lim (x -> 0) (sin(x)/x) = 1. Since the limit exists and equals the function value at x = 0, it is continuous.

Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.

Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.

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 the fact that as x approaches 0, sin(x) is approximately equal to x. This means sin(x)/x approaches 1 as x gets very close to 0.
Step 4: Therefore, we conclude that lim (x -> 0) (sin(x)/x) = 1.
Step 5: To check for continuity, we need to see if the limit we found (1) equals the function value at x = 0. The function sin(x)/x is not defined at x = 0, but we can define it to be 1 to make it continuous.
Step 6: Since the limit exists and we can define the function value at x = 0 to be equal to the limit, the function is continuous at x = 0.

Limit Evaluation – Understanding how to evaluate the limit of a function as it approaches a specific value.
Continuity – Determining if a function is continuous at a point by checking if the limit equals the function value.
Sine Function Behavior – Knowledge of the behavior of the sine function near zero and its standard limit.