Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x =

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Question: Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?

Options:

Correct Answer: Yes

Solution:

Both limits as x approaches 0 from the left and right are equal to 1, hence f(x) is continuous at x = 0.

Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?

Is the function f(x) = { e^x, x < 0; ln(x + 1), x >= 0 } continuous at x = 0?

Correct Answer: Yes, f(x) is continuous at x = 0.

Step 1: Identify the function f(x) which is defined in two parts: f(x) = e^x for x < 0 and f(x) = ln(x + 1) for x >= 0.
Step 2: To check if f(x) is continuous at x = 0, we need to find the limit of f(x) as x approaches 0 from the left (x < 0) and from the right (x >= 0).
Step 3: Calculate the limit as x approaches 0 from the left: lim (x -> 0-) f(x) = lim (x -> 0-) e^x = e^0 = 1.
Step 4: Calculate the limit as x approaches 0 from the right: lim (x -> 0+) f(x) = lim (x -> 0+) ln(x + 1) = ln(0 + 1) = ln(1) = 0.
Step 5: Compare the two limits: The limit from the left is 1 and the limit from the right is 0. Since they are not equal, f(x) is not continuous at x = 0.

Continuity of Functions – Understanding the definition of continuity at a point, which requires that the left-hand limit, right-hand limit, and the function value at that point are all equal.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, particularly at the point where the definition changes.
Limits – Calculating and comparing the limits of a function as it approaches a specific point from both sides.