What is {h→0}{lim}{{2x+3h}-{2x}}{2h} equal to?

Correct Answer is : 342x{3}{4{2x}}42x3

Correct Answer is : 342x{3}{4{2x}}42x3

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UPSC NDA I 2018 Mathematics Solved Paper

Question : 79 of 120

Marks: +1, -0

What is

\underset{h→0}{lim}\frac{\sqrt{2x+3h}-\sqrt{2x}}{2h}

equal to?

Solution:

Concept:

Use rationalization to simplify the limit expression by eliminating the square roots in the numerator.

Explanation:

We need to find

\underset{h\to 0}{\lim} \frac{\sqrt{2x+3h} - \sqrt{2x}}{2h}

Rationalize the numerator by multiplying both numerator and denominator by the conjugate

\sqrt{2x+3h} + \sqrt{2x}

\underset{h\to 0}{\lim} \frac{(\sqrt{2x+3h} - \sqrt{2x})(\sqrt{2x+3h} + \sqrt{2x})}{2h(\sqrt{2x+3h} + \sqrt{2x})}

The numerator simplifies to

(2x+3h) - (2x) = 3h

So the expression becomes

\underset{h\to 0}{\lim} \frac{3h}{2h(\sqrt{2x+3h} + \sqrt{2x})}

Cancel

h

(since

h \neq 0

\underset{h\to 0}{\lim} \frac{3}{2(\sqrt{2x+3h} + \sqrt{2x})}

Now take the limit

h \to 0

\frac{3}{2(\sqrt{2x} + \sqrt{2x})} = \frac{3}{2(2\sqrt{2x})} = \frac{3}{4\sqrt{2x}}

Answer:

\frac{3}{4\sqrt{2x}}

(Option D).

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