First slide

Introduction to limits

Question

$If f (x) = \frac{2}{x - 3}, g (x) = \frac{x - 3}{x + 4}, and h (x) = - \frac{2 (2 x + 1)}{x^{2} + x - 12}, then \lim_{x \to 3} [f (x) + g (x) + h (x)] is$

Moderate

A

$- 2$
B

$- 1$
C

$- \frac{2}{7}$
D

0

Solution

$We have \begin{array}{l} f (x) + g (x) + h (x) = \frac{x^{2} - 4 x + 17 - 4 x - 2}{x^{2} + x - 12} \\ = \frac{x^{2} - 8 x + 15}{x^{2} + x - 12} = \frac{(x - 3) (x - 5)}{(x - 3) (x + 4)} \\ ∴ \lim_{x \to 3} [f (x) + g (x) + h (x)] = \lim_{x \to 3} \frac{(x - 3) (x - 5)}{(x - 3) (x + 4)} = - \frac{2}{7} \end{array}$

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