IGNOU MST 1 2018 - Prove that

Question

Prove that $\frac{\sqrt{2 + x^{2}}-\sqrt{2-x^{2}}}{x^{2}} =\frac{1}{\sqrt{2}}$

16 Feb 2018

Answer :

Word Count : 327

We need to prove the following equation numerically: \[ \frac{\sqrt{2 + x^{2}} - \sqrt{2 - x^{2}}}{x^{2}} = \frac{1}{\sqrt{2}} \] Let's break down the steps: 1. Assume small $ x $: One way to solve this numerically is to evaluate the left-hand side for small values of $ x $, then check if it approximates $ \frac{1}{\sqrt{2}} $. 2. Choose a small value for $ x $: Let’s choose $ x = 0.1 $ (or any small value). ### Step 1: Calculate the Left-hand Side (LHS) The left-hand side of the equation is: \[ \text{LHS} = \frac{\sqrt{2 + x^2} - \sqrt{2 - x^2}}{x^2} \] For $ x = 0.1 $: - \( \sqrt{2 + (0.1)^2} = \sqrt{2.01} \approx 1.4177

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