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Magazine Chapter 3: Problem 14
Find the limits. $$\lim _{x \rightarrow+\infty} \frac{e^{3 x}}{x^{2}}$$
Short Answer
Expert verified
The limit is infinity.
Step by step solution
01
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.
02
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.
03
Apply L'H么pital's Rule
Since the form of the limit \( \frac{e^{3x}}{x^2} \) is \( \frac{\infty}{\infty} \), we can apply L'H么pital's Rule. Differentiating the numerator and the denominator gives:- Derivative of the numerator \( (e^{3x})': 3e^{3x} \)- Derivative of the denominator \( (x^2)': 2x \).
04
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'H么pital's Rule once more.
05
Differentiate Again
Differentiating once more:- The derivative of \( 3e^{3x} \) is \( 9e^{3x} \).- The derivative of \( 2x \) is \( 2 \).The new limit is:\[\lim_{x \to +\infty} \frac{9e^{3x}}{2}\]
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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 \( x \rightarrow +fty \).
L'H么pital's Rule
L'H么pital'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:
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.
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.
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:
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 \( x \to -\infty \).
