First slide

Introduction to limits

Question

If, $f (x) = \frac{3 x^{2} + ax + a + 1}{x^{2} + x - 2},$ then which of the following can be correct?

Moderate

A

$\lim_{x \to 1} f (x) exists \Rightarrow a = - 2$
B

$\lim_{x \to 2} f (x) exists \Rightarrow a = - 13$
C

$\lim_{x \to 1} f (x) = \frac{4}{3} if it exists$
D

$\lim_{x \to - 2} f (x) = \frac{1}{3} if it exists$

Solution

$\begin{array}{l} f (x) = \frac{3 x^{2} + ax + a + 1}{(x + 2) (x - 1)} \\ As x \to 1, D^{r} \to 0 . Hence x \to 1, N^{r} \to 0 . Therefore, \\ 3 + 2 a + 1 = 0 or a = - 2 \\ As x \to 2, D^{r} \to 0, Hence as x \to - 2, N^{4} \to 0 . Therefore, \\ 12 - 2 a + a + 1 = 0 or a = 13 \\ Now, \lim_{x \to 1} f (x) = \frac{\lim_{x \to 1} 3 x^{2} - 2 x - 1}{(x + 2) (x - 2)} = \lim_{x \to 1} \frac{(3 x + 1) (x - 1)}{(x + 2) (x - 1)} = \frac{4}{3} \\ Now, \lim_{x \to 2} \frac{3 x^{2} + 13 x + 14}{(x + 2) (x - 1)} = \lim_{x \to 2} \frac{(3 x + 7) (x + 2)}{(x + 2) (x - 1)} = \frac{1}{3} \end{array}$

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