A civil engineer is calculating the stress on a bridge support. The stress S (in Pascals) at a certain point is given by the function S(r) = (r^3 - 8) / (r - 2), where r is a variable related to the design parameters. What is the value of S(r) as r approaches 2?
- A12
- B0
- C4
- D8
Show answer & explanationAnswer & explanation
Correct answer: A. 12
If we substitute r = 2 directly into the function, we get (2^3 - 8) / (2 - 2) = 0/0, which is an indeterminate form. This indicates a removable discontinuity (a hole). To find the limit, factor the numerator using the difference of cubes formula (a^3 - b^3) = (a - b)(a^2 + ab + b^2). So, r^3 - 8 = (r - 2)(r^2 + 2r + 4). The function becomes S(r) = [(r - 2)(r^2 + 2r + 4)] / (r - 2). Cancel the common factor (r - 2) (assuming r ≠ 2). Then, S(r) = r^2 + 2r + 4. Now substitute r = 2 into the simplified expression: S(2) = 2^2 + 2(2) + 4 = 4 + 4 + 4 = 12.
Why the other options are wrong
- B. This is the result of 0/0, but it's an indeterminate form, not the limit.
- C. This might be 2^2, or a partial calculation.
- D. This is 2^3, the numerator term, not the limit.
Limits of Rational Functions with Holes
If direct substitution into a rational function results in the indeterminate form 0/0, it suggests a removable discontinuity (hole). To find the limit, factor the numerator and denominator, cancel common factors, and then substitute the value.
- 0/0 indicates a hole or removable discontinuity.
- Factorization helps identify common factors.
- Difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
Memory trick: Zero Over Zero? Factor, Cancel, Then Go!