Let f : (0, 1) → ℝ be the functions defined as if where n ∈ N. Let g : (0, 1) → ℝ be a function such that for all x ∈ (0, 1). then

Engineering

Mathematics

Introduction to Limit

Methods to Evaluate Limits

Properties of Definite Integral

Question

Let f : (0, 1) → ℝ be the functions defined as $f (x) = \sqrt{n}$ if $x \in [\frac{1}{n + 1}, \frac{1}{n})$ where n ∈ N. Let g : (0, 1) → ℝ be a function such that $\int_{x^{2}}^{x} \sqrt{\frac{1 - t}{t}} dt < g (x) < 2 \sqrt{x}$ for all x ∈ (0, 1). then $lim_{x \to 0} f (x) g (x)$

is equal to 3

is equal to 2

is equal to 1

does NOT exist

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$\int_{x^{2}}^{x} \sqrt{\frac{1 - t}{t}} dt \cdot \sqrt{n} \leq f (x) g (x) \leq 2 \sqrt{x} \sqrt{n}$

$∵ \int_{x^{2}}^{x} \sqrt{\frac{1 - t}{t}} dt = \sin^{- 1} \sqrt{x} + \sqrt{x} \sqrt{1 - x} - \sin^{- 1} x - x \sqrt{1 - x^{2}}$

$\Rightarrow lim_{x \to 0} (\frac{\sin^{- 1} \sqrt{x} + \sqrt{x} \sqrt{1 - x} - \sin^{- 1} x - x \sqrt{1 - x^{2}}}{\sqrt{x}} \leq f (x) g (x) \leq \frac{2 \sqrt{x}}{\sqrt{x}})$

$\Rightarrow 2 \leq lim_{x \to 0} f (x) g (x) \leq 2$

$\Rightarrow lim_{x \to 0} f (x) g (x) = 2$