Problem 48 - Calculus Early Transcendentals

Question

Using L'Hôpital's rule (Section 3.6) one can verify that $$ \lim _{x \rightarrow+\infty} \frac{e^{x}}{x}=+\infty, \quad \lim _{x \rightarrow+\infty} \frac{x}{e^{x}}=0, \quad \lim _{x \rightarrow-\infty} x e^{x}=0 $$ In these exercises: (a) Use these results, as necessary, to find the limits of \(f(x)\) as \(x \rightarrow+\infty\) and as \(x \rightarrow-\infty\). (b) Sketch a graph of \(f(x)\) and identify all relative extrema, inflection points, and asymptotes (as appropriate). Check your work with a graphing utility. $$ f(x)=x^{2} e^{2 x} $$

Step-by-Step Solution

Answer

As \(x \to \infty\), \(f(x) \to \infty\); as \(x \to -\infty\), \(f(x) \to 0\).

Step 1: Analyze the Limit as x Approaches Infinity

We need to find \(\lim_{x \rightarrow \infty} f(x)\) where \(f(x) = x^2 e^{2x}\). As \(x\) approaches infinity, the exponential term \(e^{2x}\) dominates over the polynomial term \(x^2\). Since \(\lim_{x \rightarrow \infty} \frac{x}{e^x} = 0\), it follows that \(\lim_{x \rightarrow \infty} \frac{x^2}{e^{2x}} = 0\) because \((x^2)/(e^{2x}) = (x/e^x)^2\). Then \(\lim_{x \rightarrow \infty} x^2 e^{2x} = \infty\).

Step 2: Analyze the Limit as x Approaches Negative Infinity

For \(\lim_{x \rightarrow -\infty} f(x)\), the exponential term \(e^{2x}\) approaches 0 much faster than the polynomial term grows, due to \(2x\) becoming a large negative number. Therefore, \(x^2 e^{2x} \approx 0\) as \(x^2\) doesn't change the overall effect as \(e^{2x} \to 0\), so \(\lim_{x \rightarrow -\infty} x^2 e^{2x} = 0\).

Step 3: Sketching the Graph of f(x)

The graph of \(f(x) = x^2 e^{2x}\) will have an asymptote towards \(\infty\) as \(x \to \infty\) and approach 0 as \(x \to -\infty\). There are no relative extrema or inflection points since the graph is always increasing as \(x\to \infty\) and decreasing as \(x\to -\infty\). Use a graphing utility for a more precise sketch based on these observations. However, be cautious of rounding errors in extreme ranges.

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical function that grow very quickly as their input increases. These functions have the form \( a^{x} \) where \( a \) is a constant and \( x \) is the variable. A unique feature of exponential functions is their constant growth factor, which makes them accelerate rapidly compared to polynomial functions when \( x \) gets large.
Understanding how these functions behave is essential in calculus, especially when dealing with limits and finding function behavior at both positive and negative infinities. In the exercise given, this is evident as the exponential term \( e^{2x} \) dominates over the polynomial \( x^2 \) when \( x \to +\infty \).

• The exponential term grows faster because even as a small base raised to a large power, it increases more swiftly than any polynomial power.
• This characteristic means that, for sufficiently large \( x \), \( e^{2x} \) will be much larger than \( x^2 \), leading to the integral importance of considering exponential terms in limit calculations.

Such functions are important in many fields, such as finance for compound interest calculations, in population dynamics, and natural phenomena where growth is proportional to the current value.

Limits at Infinity

Limits are a core concept in calculus used to describe the behavior of functions as they approach certain values or even infinity. When analyzing a function like \( f(x) = x^2 e^{2x} \) as \( x \to +\infty \), it's important to discern which part of the equation grows faster and how that influences the entire function.
When \( x \to +\infty \), the expression \( \frac{x^2}{e^{2x}} = \left(\frac{x}{e^x}\right)^2 \) supplements our understanding since \( \lim_{x \to \infty} \frac{x}{e^x} = 0 \). This reveals that \( x^2 e^{2x} \to \infty \) because the bottom grows much faster, leading to the overall fraction going to zero.

• As \( x \to -\infty \), the term \( e^{2x} \to 0 \) much more rapidly in the negative, dragging \( x^2 e^{2x} \) to 0.
• Recognizing these subtle behaviors is crucial for accurate predictions and sketches of functions over their entire domain.
• L'Hôpital's Rule is particularly useful when dealing with indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), which frequently occur in limit computations involving exponential functions.

By leveraging knowledge about how fast different parts of a function grow or shrink, students can predict the limits of even complex functions at infinity.

Graphical Analysis

Graphical analysis allows us to visually interpret and understand the behavior of functions, like the provided function \( f(x) = x^2 e^{2x} \). When sketching a graph, you obtain insights such as increasing and decreasing trends, asymptotes, and potential extrema.
For this function:

• As \( x \rightarrow +\infty \), the term \( e^{2x} \) ensures that the graph shoots towards infinity.
• When \( x \rightarrow -\infty \), \( e^{2x} \to 0 \) causes the function to approach zero.
• The lack of relative extrema or inflection points means the graph smoothly increases as \( x \to +\infty \) and smoothly decreases as \( x \to -\infty \), without any turns or changes in curvature that might indicate maxima, minima, or points of inflection.

It's crucial to use graphing utilities to model such functions, ensuring visual accuracy while being mindful of typical computational issues like rounding errors with extreme values. This graphical perspective is instrumental in confirming analytical conclusions drawn during limit evaluations.