Problems based on $1^\infty$ Form

When dealing with limits in calculus, we often encounter indeterminate forms. One such form is $1^\infty$, which occurs when a base that approaches 1 is raised to a power that approaches infinity. This form is indeterminate because it is not clear whether the expression should tend towards 1, since any number to the power of 0 is 1, or whether it should tend towards infinity, since 1 raised to an infinite power could be infinitely large.

Understanding $1^\infty$ Form

The $1^\infty$ form is an indeterminate form because it does not lead to a unique limit. It can result in a variety of different values depending on the particular functions involved. To evaluate limits of this form, we often use a technique involving the natural logarithm and L'Hôpital's Rule.

Evaluating $1^\infty$ Form

To evaluate a limit that results in the $1^\infty$ form, we can use the following steps:

1. Recognize the $1^\infty$ form.
2. Take the natural logarithm of the expression.
3. Use L'Hôpital's Rule to evaluate the limit of the natural logarithm of the expression.
4. Exponentiate the result to obtain the original limit.

The general formula for evaluating limits of the form $1^\infty$ is:

$$ \lim_{x \to c} \left[ f(x) \right]^{g(x)} = e^{\lim_{x \to c} g(x) \cdot (\ln(f(x)) - \ln(1))} $$

where $f(x)$ approaches 1 and $g(x)$ approaches infinity as $x$ approaches $c$.

Table of Differences and Important Points

Aspect	Description
Indeterminate Form	$1^\infty$ is an indeterminate form because it does not have a standard limit value.
Technique	Use the natural logarithm and L'Hôpital's Rule to evaluate the limit.
Natural Logarithm	Taking the natural logarithm simplifies the expression and allows for the application of L'Hôpital's Rule.
L'Hôpital's Rule	This rule is used to evaluate limits that result in $0/0$ or $\infty/\infty$ forms.
Exponentiation	After finding the limit of the natural logarithm, exponentiate to find the original limit.

Examples

Let's go through some examples to illustrate how to handle limits of the form $1^\infty$.

Example 1

Evaluate the limit:

$$ \lim_{x \to 0} (1 + x)^{\frac{1}{x}} $$

Solution:

Recognize the $1^\infty$ form:

$$ f(x) = 1 + x \to 1 \quad \text{and} \quad g(x) = \frac{1}{x} \to \infty \quad \text{as} \quad x \to 0 $$

Take the natural logarithm:

$$ \ln \left[ \lim_{x \to 0} (1 + x)^{\frac{1}{x}} \right] = \lim_{x \to 0} \frac{1}{x} \ln(1 + x) $$

Apply L'Hôpital's Rule:

$$ \lim_{x \to 0} \frac{\ln(1 + x)}{x} = \lim_{x \to 0} \frac{\frac{1}{1 + x}}{1} = 1 $$

Exponentiate:

$$ e^{\lim_{x \to 0} \frac{\ln(1 + x)}{x}} = e^1 = e $$

Therefore, the limit is $e$.

Example 2

Evaluate the limit:

$$ \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$

Solution:

Recognize the $1^\infty$ form:

$$ f(x) = 1 + \frac{1}{x} \to 1 \quad \text{and} \quad g(x) = x \to \infty \quad \text{as} \quad x \to \infty $$

Take the natural logarithm:

$$ \ln \left[ \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x \right] = \lim_{x \to \infty} x \ln\left(1 + \frac{1}{x}\right) $$

Apply L'Hôpital's Rule:

$$ \lim_{x \to \infty} \frac{\ln(1 + \frac{1}{x})}{\frac{1}{x}} = \lim_{x \to \infty} \frac{\frac{-1}{x^2(1 + \frac{1}{x})}}{\frac{-1}{x^2}} = \lim_{x \to \infty} \frac{1}{1 + \frac{1}{x}} = 1 $$

Exponentiate:

$$ e^{\lim_{x \to \infty} x \ln\left(1 + \frac{1}{x}\right)} = e^1 = e $$

Therefore, the limit is $e$.

These examples demonstrate the process of evaluating limits in the $1^\infty$ form. It's important to practice with a variety of functions to become proficient in recognizing and solving these types of limit problems.