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At x = -1:
- Left-hand limit: limₓ→₋₁⁻ f(x) = 1 (since f(x) = 1 for x ≤ -1).
- Right-hand limit: limₓ→₋₁⁺ f(x) = -(-1) = 1 (since f(x) = -x for -1 < x < 0).
- Since both one-sided limits equal 1 and f(-1) = 1, the function is continuous at x = -1.
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At x = 0:
- Left-hand limit: limₓ→₀⁻ f(x) = -(0) = 0 (since f(x) = -x for -1 < x < 0).
- Right-hand limit: limₓ→₀⁺ f(x) = -(0) = 0 (since f(x) = -x for 0 < x < 1).
- The two-sided limit exists and equals 0.
- However, f(0) = 1.
- Since limₓ→₀ f(x) = 0 ≠ f(0) = 1, there is a removable discontinuity at x = 0. It can be removed by redefining f(0) = 0.
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At x = 1:
- Left-hand limit: limₓ→₁⁻ f(x) = -(1) = -1 (since f(x) = -x for 0 < x < 1).
- Right-hand limit: limₓ→₁⁺ f(x) = 1 (since f(x) = 1 for x ≥ 1).
- Since the left-hand limit (-1) does not equal the right-hand limit (1), the two-sided limit does not exist.
- f(1) = 1, so the function is right-continuous at x = 1 but not left-continuous.
- This is a jump discontinuity, which is not removable.
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At x = -1:
- Left-hand limit: limₓ→₋₁⁻ f(x) = 0 (since f(x) = 0 for x ≤ -1).
- Right-hand limit: limₓ→₋₁⁺ f(x) = 1/(-1) = -1 (since f(x) = 1/x for 0 < |x| < 1, and near -1 from the right, |x| = -x, so x is negative and approaching -1).
- Since the left-hand limit (0) ≠ right-hand limit (-1), the two-sided limit does not exist.
- f(-1) = 0, so the function is left-continuous at x = -1 but not right-continuous.
- This is a jump discontinuity, which is not removable.
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At x = 0:
- Left-hand limit: limₓ→₀⁻ f(x) = 1/x → -∞ (as x approaches 0 from the left, 1/x becomes large negative).
- Right-hand limit: limₓ→₀⁺ f(x) = 1/x → +∞ (as x approaches 0 from the right, 1/x becomes large positive).
- Both one-sided limits are infinite and do not exist as finite numbers.
- f(0) is undefined in the given piecewise definition (the case 0 < |x| < 1 excludes x=0).
- There is an infinite discontinuity at x = 0, which is not removable.
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At x = 1:
- Left-hand limit: limₓ→₁⁻ f(x) = 1/1 = 1 (since f(x) = 1/x for 0 < |x| < 1, and approaching 1 from the left, x is positive and less than 1).
- Right-hand limit: limₓ→₁⁺ f(x) = 1 (since f(x) = 1 for x > 1).
- Both one-sided limits equal 1.
- f(1) = 0.
- Since limₓ→₁ f(x) = 1 ≠ f(1) = 0, there is a removable discontinuity at x = 1. It can be removed by redefining f(1) = 1.
Parent Tip: Review the logic above to help your child master the concept of limits and continuity practice problems.
