Limit Continuity and Differentiability 1 Question 6
7. $\lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^{4}}}-\sqrt{2}}{y^{4}}$
(2019 Main, 9 Jan I)
(a) exists and equals $\frac{1}{4 \sqrt{2}}$
(b) does not exist
(c) exists and equals $\frac{1}{2 \sqrt{2}}$
(d) exists and equals $\frac{1}{2 \sqrt{2}(\sqrt{2}+1)}$
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Answer:
Correct Answer: 7. (a)
Solution:
- $f(x)=\frac{1-x(1+|1-x|)}{|1-x|} \cos \frac{1}{1-x}$
Now, $\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{-}} \frac{1-x(1+1-x)}{1-x} \cos \frac{1}{1-x}$
$$ =\lim _{x \rightarrow 1^{-}}(1-x) \cos \frac{1}{1-x}=0 $$
and $\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1^{+}} \frac{1-x(1-1+x)}{x-1} \cos \frac{1}{1-x}$
$=\lim _{x \rightarrow 1^{+}}-(x+1) \cdot \cos \frac{1}{x+1}$, which does not exist.
