Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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Lim X → a ( X + 2 ) 3 / 2 − ( a + 2 ) 3 / 2 X − a - Mathematics

\[\lim_{x \to a} \frac{\left( x + 2 \right)^{3/2} - \left( a + 2 \right)^{3/2}}{x -  a}\]

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Solution

\[\lim_{x \to a} = \left[ \frac{\left( x + 2 \right)^\frac{3}{2} - \left( a + 2 \right)^\frac{3}{2}}{x - a} \right]\]
\[ = \lim_{x \to a} \left[ \frac{\left( x + 2 \right)^\frac{3}{2} - \left( a + 2 \right)^\frac{3}{2}}{\left( x + 2 \right) - \left( a + 2 \right)} \right]\] 

Let y = x + 2 and b = a + 2.

When x ​→ a and x + 2 ​→ a + 2. 

\[\Rightarrow\]y ​→ 

\[\lim_{y \to b} \left[ \frac{y^\frac{3}{2} - b^\frac{3}{2}}{y - b} \right]\]
\[ = \frac{3}{2} \left( b \right)^\frac{3}{2} - 1 \]
\[ = \frac{3}{2} b^\frac{1}{2} \]
\[ = \frac{3}{2} \left( a + 2 \right)^\frac{1}{2}\]

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.5 | Q 2 | Page 33
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