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Chapter 8: Problem 24

Compute the limits. $$ \lim _{x \rightarrow-\infty} \frac{x+x^{-1}}{1+\sqrt{1-x}} $$

Short Answer

Expert verified
The computed limit is \( -\infty\).

Step by step solution

01

Simplify the function

First, try to simplify the function. Notice that \(x^{-1}\) tends to zero as \(x\) tends to negative infinity. The simplified function becomes: \[ \lim _{x \rightarrow-\infty} \frac{x}{1+\sqrt{1-x}} \] because \( \lim_{x\to -\infty} x^{-1} = 0 \).
02

Apply limit properties

Next, apply the limit properties. Divide the numerator and denominator of the function by \(x\) to reduce its complexity. Therefore, the expression simplifies to: \[\lim _{x \rightarrow-\infty} \frac{1}{\frac{1}{x}+\sqrt{\frac{1}{x}-1}} \] Here, as \(x \rightarrow -\infty\), \( \frac{1}{x} \rightarrow 0 \). Thus, the expression above turns into: \[ \lim _{x \rightarrow-\infty} \frac{1}{0+0} \].
03

Evaluate final limit

Dividing any number by zero results in undefined or infinity. Here, because the denominator is approaching zero, the final limit value will be \( -\infty\).

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