SATHEE: Limit Continuity and Differentiability 1 Question 8

Limit Continuity and Differentiability 1 Question 8

9. $lim_{x \to 0} \frac{\sin (π \cos^{2} x)}{x^{2}}$ is equal to

(2014 Main)

(a) $\frac{π}{2}$

(b) 1

(d) $π$

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Answer:

Correct Answer: 9. (d)

Solution:

PLAN To make the quadratic into simple form we should eliminate radical sign.

Description of Situation As for given equation, when $a \to 0$ the equation reduces to identity in $x$ .

i.e. $a x^{2} + b x + c = 0, \forall x \in R$ or $a = b = c \to 0$

Thus, first we should make above equation independent from coefficients as 0 .

Let $a + 1 = t^{6}$ . Thus, when $a \to 0, t \to 1$ .

$\begin{aligned} ∴ & (t^{2} - 1) x^{2} + (t^{3} - 1) x + (t - 1) & = 0 \\ \Rightarrow (t - 1) (t + 1) x^{2} + (t^{2} + t + 1) x + 1 & = 0, as t \to 1 \\ 2 x^{2} + 3 x + 1 & = 0 \\ \Rightarrow & 2 x^{2} + 2 x + x + 1 & = 0 \\ \Rightarrow & (2 x + 1) (x + 1) & = 0 \\ Thus, & x & = - 1, - 1 / 2 \end{aligned}$

or $lim_{a \to 0^{+}} α (a) = - 1 / 2$

and $lim_{a \to 0^{+}} β (a) = - 1$

Limit Continuity and Differentiability 1 Question 8

9. $lim_{x \to 0} \frac{\sin (π \cos^{2} x)}{x^{2}}$ is equal to

Answer:

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Limit Continuity and Differentiability 1 Question 8

9. limx→0sin⁡(πcos2⁡x)x2 is equal to

Answer:

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9. $lim_{x \to 0} \frac{\sin (π \cos^{2} x)}{x^{2}}$ is equal to