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Understand the problem

True or False?
Over the real line,
$$\lim_{x\to \infty} log(1+ \sqrt{4+x} – \sqrt{1+x)}= log 2$$.

Source of the problem
TIFR GS 2017 MATHEMATICS ENTRANCE EXAMINATION PAPER
Real analysis
easy
Suggested Book
S.K. Mapa and Bertle and Sherbert.

Do you really need a hint? Try it first!

Since $$log$$ is a continuous function, $$\lim_{x\to \infty} log(1+ \sqrt{4+x} – \sqrt{1+x}) = log \lim_{x\to \infty}(1+ \sqrt{4+x} – \sqrt{1+x})$$. Rationalize the portion which is in the square root.
$$\log \lim_{x\to \infty}(1+ \frac{\sqrt{4+x} -\sqrt{1+x}}{\sqrt{4+x} + \sqrt{1+x}} ( \frac{\sqrt{4+x} + \sqrt{1+x}}{ \sqrt{4+x}+ \sqrt{1+x}}))$$ = $$\log \lim_{x\to \infty}(1 + \frac {3}{(\sqrt{4+x}+ \sqrt{1+x})^2})$$
Calculate the limit. You will get the value $$\log1= 0$$. But the given value of the limit is $$log2$$ Hence, the statement is false.

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