Questions: lim as x approaches -∞ of (1-x^2)/(1+x^2)

Transcript text: $\lim _{x \rightarrow-\infty} \frac{1-x^{2}}{1+x^{2}}$

Solution

Solution Steps

Step 1: Analyze the limit as $ x $ approaches $-\infty$

We are tasked with evaluating the limit: \[ \lim _{x \rightarrow-\infty} \frac{1 - x^{2}}{1 + x^{2}} \]

Step 2: Divide numerator and denominator by $ x^{2} $

To simplify the expression, divide both the numerator and the denominator by $ x^{2} $, the highest power of $ x $ in the denominator: \[ \lim _{x \rightarrow-\infty} \frac{\frac{1}{x^{2}} - 1}{\frac{1}{x^{2}} + 1} \]

Step 3: Evaluate the limit of each term

As $ x \rightarrow -\infty $, $ \frac{1}{x^{2}} \rightarrow 0 $. Substitute this into the expression: \[ \frac{0 - 1}{0 + 1} = \frac{-1}{1} = -1 \]

Thus, the limit evaluates to $-1$.

Final Answer

$\boxed{-1}$

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