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Question
\[\lim_{x \to a} \frac{\left( x + 2 \right)^{5/2} - \left( a + 2 \right)^{5/2}}{x - a}\]
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Solution
\[\lim_{x \to a} \left[ \frac{\left( x + 2 \right)^\frac{5}{2} - \left( a + 2 \right)^\frac{5}{2}}{x - a} \right]\]
\[ = \lim_{x \to a} \left[ \frac{\left( x + 2 \right)^\frac{5}{2} - \left( a + 2 \right)^\frac{5}{2}}{\left( x + 2 \right) - \left( a + 2 \right)} \right]\]
Let y = x + 2 and b = a + 2.
When x → a, then x + 2 → a + 2.
⇒ y → b
\[\lim_{y \to b} \left[ \frac{y^\frac{5}{2} - b^\frac{5}{2}}{y - b} \right]\]
\[ = \frac{5}{2} \left( b \right)^\frac{5}{2} - 1 \]
\[ = \frac{5}{2} b^\frac{3}{2} \]
\[ = \frac{5}{2} \left( a + 2 \right)^\frac{3}{2}\]
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