One Problem at a Time

Limit and The Fundamental Theorem of Calculus

Problem 186: Find

(1)   \begin{equation*} \lim_{x \to \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} \frac{dt}{\sqrt{t}}.\end{equation*}

Solution: First, we will find the integral by using the Fundamental Theorem of Calculus. Secondly, we will find the limit. That is,

(2)   \begin{align*} \lim_{x \to \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} \frac{dt}{\sqrt{t}} & = \lim_{x \to \infty} \frac{1}{\sqrt{x}}\int_{1}^{x} t^{-1/2} dt \\ & = \lim_{x \to \infty} \frac{1}{\sqrt{x}} \cdot \left(2\sqrt{t} \right)\bigg|_{1}^{x} \\ & = \lim_{x \to \infty} \frac{1}{\sqrt{x}} \cdot \left( 2\sqrt{x} - 2\sqrt{1} \right) \\ & = \lim_{x \to \infty} 2 - \frac{2}{\sqrt{x}} \\ & = 2. \end{align*}

That is,

(3)   \begin{equation*} \lim_{x \to \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} \frac{dt}{\sqrt{t}} = 2. \end{equation*}

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