Question: Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
Options:
Yes
No
Only left continuous
Only right continuous
Correct Answer: Yes
Solution:
Both limits as x approaches 1 from the left and right are equal to 2, hence f(x) is continuous at x = 1.
Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
Practice Questions
Q1
Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
Yes
No
Only left continuous
Only right continuous
Questions & Step-by-Step Solutions
Is the function f(x) = { x^3, x < 1; 2x + 1, x >= 1 } continuous at x = 1?
Correct Answer: Yes, f(x) is continuous at x = 1.
Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^3 when x is less than 1, and f(x) = 2x + 1 when x is greater than or equal to 1.
Step 2: Find the limit of f(x) as x approaches 1 from the left (x < 1). This means we use the first part of the function: f(x) = x^3. So, calculate the limit: lim (x -> 1-) f(x) = 1^3 = 1.
Step 3: Find the limit of f(x) as x approaches 1 from the right (x >= 1). This means we use the second part of the function: f(x) = 2x + 1. So, calculate the limit: lim (x -> 1+) f(x) = 2(1) + 1 = 3.
Step 4: Compare the two limits from Step 2 and Step 3. The left limit is 1 and the right limit is 3. Since they are not equal, the function is not continuous at x = 1.
Continuity of Functions – Understanding the definition of continuity at a point, which requires that the limit from the left equals the limit from the right and both equal the function's value at that point.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value, particularly at the point where the definition changes.
Limit Evaluation – Calculating the left-hand and right-hand limits to determine continuity.
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