Limits Continuity And Differentiability Question 2
Question: Let f’(x) be continuous at x = 0 and f"(0) = 4. Then value of $ \underset{x\to 0}{\mathop{\lim }}\frac{2f(x)-3f(2x)+f(4x)}{x^{2}} $ is
Options:
A) 12
B) 10
C) 6
D) 4
Show Answer
Answer:
Correct Answer: A
Solution:
Given f?? (x) is continuous at x = 0 $ =\underset{x\to 0}{\mathop{\lim }}f’’(x)=f’’(0)=4 $
Now, $ =\underset{x\to 0}{\mathop{\lim }}\frac{2f(x)-3f(2x)+f(4x)}{x^{2}}[ \frac{0}{0}form ] $
$ =\underset{x\to 0}{\mathop{\lim }}\frac{2f’(x)-6f’(2x)+4f’(4x)}{2x}[ \frac{0}{0}form ] $
$ =\underset{x\to 0}{\mathop{\lim }}\frac{2f’’(x)-12f’’(x)+16f’’(4x)}{2} $
[Using L’hospital’s Rule successively] $ =\frac{2f’’(0)-12f’’(0)+16f’’(0)}{2}=12 $
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