Problem 14 - Calculus - AP Edition

Question

Find the limits. $$\lim _{x \rightarrow+\infty} \frac{e^{3 x}}{x^{2}}$$

Step-by-Step Solution

Answer

The limit is infinity.

Step 1: Identify the Limit

The limit to be solved is \( \lim_{x \to +\infty} \frac{e^{3x}}{x^2} \). This expression is a ratio of an exponential function \( e^{3x} \) in the numerator and a polynomial \( x^2 \) in the denominator.

Step 2: Analyze the Exponential Growth

Exponential functions grow faster than polynomial functions as \( x \) approaches infinity. In this case, \( e^{3x} \) grows significantly faster than \( x^2 \) since any exponential term will eventually overpower any polynomial term. This suggests that the limit may approach infinity.

Step 3: Apply L'Hôpital's Rule

Since the form of the limit \( \frac{e^{3x}}{x^2} \) is \( \frac{\infty}{\infty} \), we can apply L'Hôpital's Rule. Differentiating the numerator and the denominator gives:- Derivative of the numerator \( (e^{3x})': 3e^{3x} \)- Derivative of the denominator \( (x^2)': 2x \).

Step 4: Evaluate the New Limit

Re-apply the limit to the new expression:\[\lim_{x \to +\infty} \frac{3e^{3x}}{2x}\]This derivative form still results in \( \frac{\infty}{\infty} \), so we apply L'Hôpital's Rule once more.

Step 5: Differentiate Again

Differentiating once more:- The derivative of \( 3e^{3x} \) is \( 9e^{3x} \).- The derivative of \( 2x \) is \( 2 \).The new limit is:\[\lim_{x \to +\infty} \frac{9e^{3x}}{2}\]

Step 6: Conclude the Limit Evaluation

The expression \( \frac{9e^{3x}}{2} \) clearly approaches infinity as \( x \to +\infty \) because \( e^{3x} \) dominates the expression. Thus, the original limit also approaches infinity.

Key Concepts

Exponential Growth

Exponential growth refers to the increase in a quantity according to an exponential function. Exponential functions, such as \( e^{3x} \), grow at a rate that is proportional to their current value, causing a rapid increase as \( x \) becomes large. This can be illustrated in contrast with polynomial functions, for example, \( x^2 \).

The key characteristics of exponential growth are:

• Rapid increase: As \( x \) increases, \( e^{3x} \) rapidly approaches very large values.
• Outpaces polynomial growth: No matter how steep the slope of a polynomial function, an exponential function \( e^{3x} \) will eventually grow much faster as \( x \rightarrow +fty \).

These properties are crucial in limit problems involving fractions of exponential and polynomial functions, where exponential functions in the numerator indicate limits that tend towards infinity.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus to find the limit of indeterminate forms like \( \frac{\infty}{\infty} \). It states that:

To find the limit of a fraction, where both numerator and denominator approach \( \infty \), differentiate each:

• If \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\infty}{\infty} \)
• Then, \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \) if this limit exists.

For the given problem, L'Hôpital's Rule was applied twice:

1. Differentiate \( e^{3x} \) and \( x^2 \) to get \( \frac{3e^{3x}}{2x} \), which remains an indeterminate form.
2. Differentiate again to derive \( \frac{9e^{3x}}{2} \), which approaches infinity as \( x \rightarrow +\infty \).

This approach simplifies evaluation of complex limits, especially when exponential terms are involved.

Polynomial Functions

Polynomial functions are algebraic expressions involving variables raised to whole-number exponents, such as \( x^2 \). Each term is composed of a coefficient and a variable raised to an exponent.

Polynomials are characterized by:

• The degree: The highest power exponent in the expression (e.g., 2 in \( x^2 \)).
• Leading term: The term with the highest exponent, it primarily determines the function's behavior as \( x \to \infty \) or \( x \to -\infty \).

In the limit problem, the simple polynomial \( x^2 \) grows steadily, but not as fast as the exponential function \( e^{3x} \), leading the overall limit to reflect the faster growing component. When comparing exponential and polynomial growth, exponential functions usually prevail, pushing limits toward infinity.