Use Continuity to evaluate the limit, \(\lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )\)

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Solution:

Given

\(\lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )\)

Clearly \(\sqrt{x}\) is continuous for \(x \geq 0\) and \(\sqrt{(5+x)}\) is continuous for

\((5 + x) \geq 0 \enspace \Rightarrow \enspace x \geq -5\)

Therefore,

\(\frac{5 + \sqrt{x}}{\sqrt{5 + x}}\)             … (ii)

is continuous for \(x \geq 0\). So, Limit of (ii) exists. And,

\( = \lim_{x \to 4} \left ( \frac{5 + \sqrt{x}}{\sqrt{5 + x}} \right )\)

\( = \frac{5 + \sqrt{x}}{\sqrt{5 + x}}\)

\( = \frac{5 + 2}{3} = \frac{7}{3}\)

 

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