Evaluate:- \(\lim\limits_{h \to a}\frac{(x+2)^{3/2}-(a+2)^{3/2}}{x-

1.	Evaluate:- \(\lim\limits_{h \to a}\frac{(x+2)^{3/2}-(a+2)^{3/2}}{x-a}\)
Answer» Let x + 2 = y, then y → a + 2 as x → a ∴ \(\lim\limits_{h \to a}\frac{(x+2)^{3/2}-(a+2)^{3/2}}{x-a}\) \(=\lim\limits_{h \to a}\frac{y^{3/2}-(a+2)^{3/2}}{y-(a+2)}\) [∵ x + 2 = y ⇒ x = y − 2] \(=\frac{3}{2}(a+2)^{\frac{3}{2}-1}\)^_-1 \(=\frac{3}{2}(a+2)^\frac{1}{2}\) [∵ \(\lim\limits_{h \to a}\frac{x^n-a^n}{x-a}=na^{n-1}\) ]

Evaluate:- \(\lim\limits_{h \to a}\frac{(x+2)^{3/2}-(a+2)^{3/2}}{x-a}\)

Answer»

Let x + 2 = y,

then y → a + 2 as x → a

∴ \(\lim\limits_{h \to a}\frac{(x+2)^{3/2}-(a+2)^{3/2}}{x-a}\)

\(=\lim\limits_{h \to a}\frac{y^{3/2}-(a+2)^{3/2}}{y-(a+2)}\) [∵ x + 2 = y ⇒ x = y − 2]

\(=\frac{3}{2}(a+2)^{\frac{3}{2}-1}\)^_-1

\(=\frac{3}{2}(a+2)^\frac{1}{2}\)

[∵ \(\lim\limits_{h \to a}\frac{x^n-a^n}{x-a}=na^{n-1}\) ]

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