Question: Is the function f(x) = sqrt(x) continuous at x = 0?
Options:
Yes
No
Only from the right
Only from the left
Correct Answer: Yes
Solution:
The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.
Is the function f(x) = sqrt(x) continuous at x = 0?
Practice Questions
Q1
Is the function f(x) = sqrt(x) continuous at x = 0?
Yes
No
Only from the right
Only from the left
Questions & Step-by-Step Solutions
Is the function f(x) = sqrt(x) continuous at x = 0?
Step 1: Understand what it means for a function to be continuous at a point. A function is continuous at a point if three conditions are met: the function is defined at that point, the limit exists at that point, and the limit equals the function's value at that point.
Step 2: Check if the function f(x) = sqrt(x) is defined at x = 0. Calculate f(0): f(0) = sqrt(0) = 0. So, the function is defined at x = 0.
Step 3: Find the limit of f(x) as x approaches 0. Calculate the limit: lim (x -> 0) f(x) = lim (x -> 0) sqrt(x) = sqrt(0) = 0.
Step 4: Compare the limit with the function's value at x = 0. We found that lim (x -> 0) f(x) = 0 and f(0) = 0.
Step 5: Since the limit exists, the function is defined at x = 0, and the limit equals the function's value, we conclude that f(x) = sqrt(x) is continuous at x = 0.
Continuity – The concept of continuity at a point involves checking if the function is defined at that point, if the limit exists, and if the limit equals the function's value at that point.
Square Root Function – Understanding the properties of the square root function, particularly its domain and behavior near zero.
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