If \textrm{ }\textrm{ } f \left(\right. x \left.\right) = \left{\right. \frac{sin \left(\right. a + 2 \left.\right) x + sinx}{x} & ; & x < 0 \\ b & ; & x = 0 \\ \frac{\left(\left(\right. x + 3 x^{2} \left.\right)\right)^{1 / 3} - x^{1 / 3}}{x^{4 / 3}} & ; & x > 0 is continuous at x = 0, theen a + 2b is equal to :

Engineering

Mathematics

Theorems on Continuity

Question

$If f (x) = {\begin{matrix} \frac{\sin (a + 2) x + sinx}{x} & ; & x < 0 \\ b & ; & x = 0 \\ \frac{{(x + 3 x^{2})}^{1 / 3} - x^{1 / 3}}{x^{4 / 3}} & ; & x > 0 \end{matrix}$

is continuous at x = 0, theen a + 2b is equal to :

– 2

– 1

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LHL = a + 3

f(0) = b

RHL = $\lim_{h \to 0} (\frac{{(1 + 3 h)}^{\frac{1}{3}} - 1}{h}) = 1$

∴ a = – 2

b = 1

∴ a + 2b = 0