The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous a
Practice Questions
Q1
The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
Yes
No
Only left continuous
Only right continuous
Questions & Step-by-Step Solutions
The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
Step 1: Identify the function f(x) which is defined in three parts: f(x) = x^2 for x < 0, f(x) = 2 for x = 0, and f(x) = x + 1 for x > 0.
Step 2: To check if the function is continuous at x = 0, we need to find the left-hand limit (lim x→0-) and the right-hand limit (lim x→0+).
Step 3: Calculate the left-hand limit: lim x→0- f(x) means we look at values of x that are slightly less than 0. For these values, f(x) = x^2. So, lim x→0- f(x) = 0^2 = 0.
Step 4: Calculate the right-hand limit: lim x→0+ f(x) means we look at values of x that are slightly greater than 0. For these values, f(x) = x + 1. So, lim x→0+ f(x) = 0 + 1 = 1.
Step 5: Compare the left-hand limit and the right-hand limit. We found that lim x→0- f(x) = 0 and lim x→0+ f(x) = 1.
Step 6: Since the left-hand limit (0) is not equal to the right-hand limit (1), the function is discontinuous at x = 0.
