- Introduce the substitution \(u = x + 8\), so that as \(x \rightarrow -8^+\), \(u \rightarrow 0^+\).
- Combine the terms into a single fraction to get \(\frac{1 - \sqrt{u}}{u}\).
- Observe that as \(u \rightarrow 0^+\), the numerator approaches 1 and the denominator approaches 0 from the positive side, leading to a value of \(\infty\).
- Conclude that the limit is \(\boxed{\infty}\).
Explanation
- Problem Analysis and Substitution
We are tasked with evaluating the limit:
\[ L = \lim_{x \rightarrow -8^+} \left( \frac{1}{x+8} - \frac{1}{\sqrt{x+8}} \right) \]
To simplify the expression, let's introduce a substitution. Let \(u = x+8\). As \(x\) approaches \(-8\) from the right (\(x \rightarrow -8^+\)), the variable \(u\) approaches \(0\) from the right (\(u \rightarrow 0^+\)). The limit can then be rewritten in terms of \(u\):
\[ L = \lim_{u \rightarrow 0^+} \left( \frac{1}{u} - \frac{1}{\sqrt{u}} \right) \]
- Algebraic Simplification
Now, let's combine the two terms into a single fraction to better analyze the behavior of the expression:
\[ \frac{1}{u} - \frac{1}{\sqrt{u}} = \frac{1 - \sqrt{u}}{u} \]
The expression becomes:
\[ L = \lim_{u \rightarrow 0^+} \frac{1 - \sqrt{u}}{u} \]
Note that as \(u \rightarrow 0^+\), the numerator \(1 - \sqrt{u}\) approaches \(1 - 0 = 1\), while the denominator \(u\) approaches \(0\) from the positive side.
- Evaluating the Limit
We examine the limit \(\lim_{u \rightarrow 0^+} \frac{1 - \sqrt{u}}{u}\). As established, the numerator approaches \(1\) and the denominator is a small positive value approaching \(0\). The quotient of a value approaching a positive constant and a value approaching \(0^+\) is positive infinity.
It is important to note that l'Hôpital's Rule is not applicable here because the limit is not of an indeterminate form such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Instead, it is of the form \(\frac{1}{0^+}\), which directly evaluates to \(\infty\).
- Final Conclusion
Based on our analysis of the behavior of the expression as \(u\) approaches \(0^+\), we conclude that the limit grows without bound. Thus, the limit is \(\infty\).
\[ \lim_{x \rightarrow -8^+} \left( \frac{1}{x+8} - \frac{1}{\sqrt{x+8}} \right) = \boxed{\infty} \]
Examples
In engineering, understanding how a function behaves near its singularities is essential. For instance, consider the electrical potential near a point charge or the structural stress near a sharp corner. These scenarios often involve mathematical expressions with denominators that approach zero. Just as we analyzed this limit, engineers determine whether these values grow infinitely (representing a physical singularity) or approach a finite, stable value, which helps in designing systems that can withstand extreme conditions or behave safely near points of failure.