Digital SATMath: Advanced MathHard
A scientist is studying a chemical reaction where the rate of reaction R (in mol/s) is given by R(x) = (x^2 - 9) / (x - 3), where x is the concentration of a reactant in M. What is the value of R(x) as x approaches 3?
- A0
- BUndefined
- C3
- D6
Show answer & explanationAnswer & explanation
Correct answer: D. 6
If we directly substitute x=3, we get (3^2 - 9) / (3 - 3) = 0/0, which is an indeterminate form. We can simplify the expression by factoring the numerator: x^2 - 9 = (x - 3)(x + 3). So, R(x) = (x - 3)(x + 3) / (x - 3). For x ≠ 3, R(x) simplifies to x + 3. As x approaches 3, the value of R(x) approaches 3 + 3 = 6.
Why the other options are wrong
- A. This would be the case if the numerator was 0 and the denominator was non-zero.
- B. While the function is undefined at x=3, its limit as x approaches 3 is a specific value.
- C. This is simply the value of x approaching, not the function's limit.
Limits of Rational Functions with Holes
If direct substitution into a rational function results in the indeterminate form 0/0, it often indicates a 'hole' in the graph. The limit can be found by factoring and canceling common factors.
- 0/0 is an indeterminate form, suggesting further analysis is needed.
- Factoring the numerator and denominator can reveal common factors.
- Canceling common factors simplifies the expression, allowing direct substitution for the limit.
- A hole exists at the x-value where the canceled factor was zero.
Memory trick: 0/0 means 'Go Go' simplify!