Compute \(\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}\)
2019
Compute \(\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}\)
- A.
1
- B.
53/12
- C.
108/7
- D.
Limit does not exist
Attempted by 68 students.
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Correct answer: C
Factor and simplify to remove the indeterminate form.
Factor the numerator: x^4 - 81 = (x^2 - 9)(x^2 + 9) = (x - 3)(x + 3)(x^2 + 9).
Factor the denominator: 2x^2 - 5x - 3 = (2x + 1)(x - 3).
Cancel the common factor (x - 3) for x ≠ 3 to get the simplified expression (x + 3)(x^2 + 9)/(2x + 1).
Evaluate the limit by substitution: at x = 3 the expression becomes (3 + 3)(9 + 9)/(2·3 + 1) = 6·18/7 = 108/7.
Therefore the limit equals 108/7.