The Two-Sided Limit at a Piecewise Junction
A student is analyzing the piecewise function f defined by f(x) = (x^2 - 9)/(x - 3) for x < 3, f(x) = k for x = 3, and f(x) = (x - 3)/(sqrt(x+1) - 2) for x > 3. A classmate claims that because the left branch simplifies to x + 3 and the right branch is a rationalized-looking fraction, the two one-sided limits must both equal 6, so setting k = 6 makes f continuous at x = 3. The student is skeptical and wants to verify each side independently before committing to a value of k. The teacher has warned that 'simplifying first' without checking the domain of the simplified expression is a common trap.
1. Compute the left-hand limit and the right-hand limit of f at x = 3 separately. Does the classmate's claim that both equal 6 hold?
Read the worked answer
Left-hand limit (x → 3⁻): For x < 3, f(x) = (x² − 9)/(x − 3) = (x−3)(x+3)/(x−3) = x + 3, valid since x ≠ 3. So lim(x→3⁻) f(x) = 3 + 3 = 6. This part of the claim is correct.
Right-hand limit (x → 3⁺): For x > 3, f(x) = (x − 3)/(√(x+1) − 2). Direct substitution gives 0/0, so multiply by the conjugate:
f(x) = (x − 3)(√(x+1) + 2) / ((x+1) − 4) = (x − 3)(√(x+1) + 2)/(x − 3) = √(x+1) + 2.
Then lim(x→3⁺) f(x) = √4 + 2 = 4.
So the one-sided limits disagree: 6 ≠ 4. The classmate's claim fails on the right side.
2. Based on your limits, what happens to f at x = 3 for any choice of k? Classify the discontinuity and explain whether any value of k could make f continuous at 3.
Read the worked answer
Since lim(x→3⁻) f(x) = 6 and lim(x→3⁺) f(x) = 4, the two-sided limit lim(x→3) f(x) does not exist - the one-sided limits are finite but unequal.
Because the two-sided limit fails to exist, no value of k can make f continuous at x = 3. Continuity requires lim(x→3) f(x) = f(3) = k, but the left side of that equation is undefined (DNE).
This is a jump discontinuity (a type of nonremovable discontinuity): both one-sided limits exist and are finite, but they differ. Assigning k = 6 or k = 4 would only match one branch; the gap between 4 and 6 is irreducible by redefining a single point. The student's skepticism was justified.