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Problem 321

# Find $\lim _{x \rightarrow \infty}\left[x \sqrt{\left(x^{2}+1\right)}-x^{2}\right]$

Expert verified
The short answer to the limit question is: $$\lim_{x \rightarrow \infty}\left[x\sqrt{x^2+1}-x^{2}\right] = 0$$.
See the step by step solution

## Step 1: Simplify the Expression inside the Square Root

First, let's simplify the expression inside the square root: $$\sqrt{x^2+1}$$ Notice that the leading term of the polynomial is x^2, so we can factor this out of the expression: $$\sqrt{x^2(1 + \frac{1}{x^2})}$$ Now, the expression inside the square root simplifies to: $$\sqrt{x^2} \cdot \sqrt{1 + \frac{1}{x^2}} = x \cdot \sqrt{1 + \frac{1}{x^2}}$$ Our expression becomes: $$\lim_{x \rightarrow \infty}\left[x(x \cdot \sqrt{1 + \frac{1}{x^2}})-x^{2}\right]$$

## Step 2: Factor out the x Term

We can factor out the x term from the expression: $$\lim _{x \rightarrow \infty}\left[x\sqrt{1 + \frac{1}{x^2}} \cdot x-x^{2}\right] = \lim _{x \rightarrow \infty} x\left(\sqrt{1 + \frac{1}{x^2}} - 1\right)$$

## Step 3: Substitute u for 1/x&

In order to simplify the expression further, we can let $$u = \frac{1}{x}$$ (since x approaches infinity, u approaches 0) Now, our expression becomes: $$\lim_{u \rightarrow 0} \frac{1}{u}\left(\sqrt{1 + u^2} - 1\right)$$

## Step 4: Rationalize the Denominator

In order to deal with the square root, we can rationalize the denominator: Multiply both the numerator and the denominator by $$\sqrt{1 + u^2} + 1$$ to get: $$\lim_{u \rightarrow 0} \frac{(\sqrt{1 + u^2} - 1)(\sqrt{1 + u^2} + 1)}{u(\sqrt{1 + u^2} + 1)}$$

## Step 5: Simplify the Expression

Now, we can simplify the expression by multiplying the numerator and denominator expressions: $$\lim_{u \rightarrow 0} \frac{(1 + u^2) - 1}{u(\sqrt{1 + u^2} + 1)} = \lim_{u \rightarrow 0} \frac{u^2}{u(\sqrt{1 + u^2} + 1)}$$ Now, we can cancel out the u term in the numerator and denominator: $$\lim_{u \rightarrow 0} \frac{u}{\sqrt{1 + u^2} + 1}$$

## Step 6: Find the Limit as u Approaches 0

Now, we can find the limit as u approaches 0: $$\lim_{u \rightarrow 0} \frac{u}{\sqrt{1 + u^2} + 1} = \frac{0}{\sqrt{1 + 0^2} + 1} = \frac{0}{1 + 1} = 0$$ So, the limit of the given expression as x approaches infinity is 0.

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