The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
Practice Questions
1 question
Q1
The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
Yes
No
Only from the right
Only from the left
The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Questions & Step-by-step Solutions
1 item
Q
Q: The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
Solution: The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Steps: 5
Step 1: Understand the function f(x). It is defined as f(x) = 1/x when x is not equal to 0, and f(0) = 0 when x is equal to 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.
Step 3: Calculate the limit: lim (x -> 0) f(x) = lim (x -> 0) (1/x). As x gets closer to 0, 1/x becomes very large (positive or negative depending on the direction).
Step 4: Since the limit does not approach a specific number (it goes to infinity), we conclude that lim (x -> 0) f(x) does not equal f(0).
Step 5: Since the limit as x approaches 0 is not equal to f(0), we say that f(x) is not continuous at x = 0.
