Let f"(x) be continuous at x $$=$$ $$0$$.If limx→$$0$$ $$2f(x)$$−$$3af(2x)+bf(8x)sin2$$⁡x exists and $$f(0)$$≠$$0$$, f'(0)≠$$0$$, then the value of $$3a/b$$ is $$______$$.

we have $$L=limx$$→$$0$$ $$2f(x)$$−$$3af(2x)+bf(8x)sin2$$⁡xFor the limit to exist, we $$have2f(0)$$−$$3af(0)+bf(0)=0$$ or $$3a$$−$$b=2$$ [∵$$f(0)$$≠$$0$$, given ]......(1) or $$L=limx$$→$$0$$ $$2f$$′(x)−$$6af$$′$$(2x)+8bf$$′$$(8x)2x2-6a+8b=03a-4b=1a=79$$, $$b=13$$

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Let f"(x) be continuous at x = 0.If $lim_{x \to 0} \frac{2 f (x) - 3 af (2 x) + bf (8 x)}{\sin^{2} x}$ exists and $f (0) \neq 0,$ $f' (0) \neq 0,$ then the value of 3a/b is ______.

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Solution

we have
$L = lim_{x \to 0} \frac{2 f (x) - 3 af (2 x) + bf (8 x)}{\sin^{2} x}$
For the limit to exist, we have
$\begin{array}{ll} 2 f (0) - 3 af (0) + bf (0) = 0 \\ or 3 a - b = 2 [∵ f (0) \neq 0, given] . . . . . . (1) \\ or L = lim_{x \to 0} \frac{2 f^{'} (x) - 6 {af}^{'} (2 x) + 8 {bf}^{'} (8 x)}{2 x} \\ 2 - 6 a + 8 b = 0 \\ 3 a - 4 b = 1 \\ a = \frac{7}{9}, b = \frac{1}{3} \end{array}$

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Column -I	Column -II
A. If $L = lim_{x \to - 1} \frac{\sqrt[3]{(7 - x)} - 2}{(x + 1)}$ , then 12L =	P. -2
B. If $L = lim_{x \to π / 4} \frac{\tan^{3} x - \tan x}{\cos (x + \frac{π}{4})}$ then -L/4=	Q. 2
C. If $L = lim_{x \to 1} \frac{(2 x - 3) (\sqrt{x} - 1)}{2 x^{2} + x - 3}$ , then 20L=	R. 1
D. If $L = lim_{x \to \infty} \frac{\log x^{n} - [x]}{[x]}$ , where $n \in N$ ([x] denotes greatest integer less than or equal to x),then -2L=	S. -1

Introduction to limits

Remember concepts with our Masterclasses.

Let f"(x) be continuous at x = 0.If limx→0 2f(x)−3af(2x)+bf(8x)sin2⁡x exists and f(0)≠0, f'(0)≠0, then the value of 3a/b is ______.

we have L=limx→0 2f(x)−3af(2x)+bf(8x)sin2⁡xFor the limit to exist, we have2f(0)−3af(0)+bf(0)=0 or 3a−b=2 [∵f(0)≠0, given ]......(1) or L=limx→0 2f′(x)−6af′(2x)+8bf′(8x)2x2-6a+8b=03a-4b=1a=79, b=13

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Let f"(x) be continuous at x = 0.If $lim_{x \to 0} \frac{2 f (x) - 3 af (2 x) + bf (8 x)}{\sin^{2} x}$ exists and $f (0) \neq 0,$ $f' (0) \neq 0,$ then the value of 3a/b is ______.