f(x)=sinax2x2,x≠034+14a,x=0 for what values of a, f(x) is continuous at x = 0

x→0Lt f(x)=f(0) x→0Lt sinax2x2=34+14a x→0Lt sinax2ax2×a=34+14a ⇒a=34+14a⇒4a2-3a-1=0⇒a=−1/4, 1

f(x)=sinax2x2,x≠034+14a,x=0 for what values of a, f(x) is continuous at x = 0

x→0Lt f(x)=f(0) x→0Lt sinax2x2=34+14a x→0Lt sinax2ax2×a=34+14a ⇒a=34+14a⇒4a2-3a-1=0⇒a=−1/4, 1

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$f (x) = \{\begin{cases} \begin{matrix} \frac{\sin a x^{2}}{x^{2}}, & x \neq 0 \\ \frac{3}{4} + \frac{1}{4 a}, & x = 0 \end{matrix} \end{cases}$ for what values of a, f(x) is continuous at x = 0

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$\frac{1}{4}, - 1$

$\frac{1}{4}, 1$

$1, - \frac{1}{4}$

$- 1, - \frac{1}{4}$

answer is C.

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Detailed Solution

$_{x \to 0}^{L t} f (x) = f (0)$

$_{x \to 0}^{L t} \frac{\sin a x^{2}}{x^{2}} = \frac{3}{4} + \frac{1}{4 a}$

$_{x \to 0}^{L t} \frac{\sin a x^{2}}{a x^{2}} \times a = \frac{3}{4} + \frac{1}{4 a}$ $\Rightarrow a = \frac{3}{4} + \frac{1}{4 a}$

$\Rightarrow 4 a^{2} - 3 a - 1 = 0 \Rightarrow$ a= $- 1 / 4, 1$

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f(x)=sinax2x2,x≠034+14a,x=0 for what values of a, f(x) is continuous at x = 0

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$f (x) = \{\begin{cases} \begin{matrix} \frac{\sin a x^{2}}{x^{2}}, & x \neq 0 \\ \frac{3}{4} + \frac{1}{4 a}, & x = 0 \end{matrix} \end{cases}$ for what values of a, f(x) is continuous at x = 0