Limits and Continuity


Graph the function


1 , x ≤ ―1
―x , ―1 < x < 0
ƒ(x) = { 1 , x = 0 ,
―x , 0 < x < 1
1 , x ≥ 1


Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = ―1 , 0 , and 1. Are any of the discontinuities removable? Explain.

Verified step by step guidance

1. Step 1: Identify the piecewise function and its components. The function ƒ(x) is defined as follows: ƒ(x) = 1 for x ≤ -1, ƒ(x) = -1/x for -1 < x < 0, ƒ(x) = 1 for x = 0, ƒ(x) = -x for 0 < x < 1, and ƒ(x) = 1 for x ≥ 1.
2. Step 2: Analyze the limits and one-sided limits at x = -1. Evaluate the left-hand limit as x approaches -1 from the left (x → -1⁻) and the right-hand limit as x approaches -1 from the right (x → -1⁺). Compare these limits to the value of the function at x = -1 to determine continuity.
3. Step 3: Analyze the limits and one-sided limits at x = 0. Evaluate the left-hand limit as x approaches 0 from the left (x → 0⁻) and the right-hand limit as x approaches 0 from the right (x → 0⁺). Compare these limits to the value of the function at x = 0 to determine continuity.
4. Step 4: Analyze the limits and one-sided limits at x = 1. Evaluate the left-hand limit as x approaches 1 from the left (x → 1⁻) and the right-hand limit as x approaches 1 from the right (x → 1⁺). Compare these limits to the value of the function at x = 1 to determine continuity.
5. Step 5: Discuss the types of discontinuities at x = -1, 0, and 1. Determine if any discontinuities are removable by checking if redefining the function at these points can make the function continuous. A discontinuity is removable if the limit exists at that point, but the function value is different or undefined.

Key Concepts

Limits

Limits describe the behavior of a function as the input approaches a certain value. For the function ƒ(x), understanding limits at x = -1, 0, and 1 involves evaluating the function's value as x approaches these points from both sides. This helps determine if the function approaches a specific value or diverges, which is crucial for analyzing continuity and discontinuities.

One-Sided Limits

One-sided limits consider the behavior of a function as the input approaches a specific value from one side only, either from the left (x → c⁻) or the right (x → c⁺). For ƒ(x), examining one-sided limits at x = -1, 0, and 1 helps identify potential discontinuities and whether the function behaves differently when approaching these points from different directions.

Continuity and Discontinuity

A function is continuous at a point if the limit exists and equals the function's value at that point. Discontinuities occur when this condition fails. For ƒ(x), analyzing continuity at x = -1, 0, and 1 involves checking if the limits and function values match. Removable discontinuities can be fixed by redefining the function at the point, while non-removable ones indicate a more fundamental break in the function's behavior.