First slide

Continuity

Question

$\begin{array}{l} If \\ f (x) = \{\begin{cases} \frac{\sin (a + 2) x + \sin x}{x}; & x < 0 \\ b & ; x = 0 \\ \frac{{(x + 3 x^{2})}^{1 / 3} - x^{1 / 3}}{x^{4 / 3}}; x > 0 \end{cases} \end{array}$
$is continuous at x = 0, then a + 2 b is equal to:$

Moderate

A

2
B

0
C

1
D

-1

Solution

$\begin{array}{l} We have left hand limit as a + 3, f (0) = b \\ and right-hand limit is \lim_{h \to 0} (\frac{(1 + 3 h)^{\frac{1}{3}} - 1}{h}) = 1 \end{array}$
For it to be continuous, left hand limit is equal to the right-hand limit and it is also equal to $f (0)$
$So, we get a = - 2, b = 1 and a + 2 b = 0$

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