Problem 47 - 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 \( x \to +\infty \), \( f(x) \to 0 \). As \( x \to -\infty \), \( f(x) \to +\infty \). Sketch has horizontal asymptote at \( y=0 \).

Step 1: Understanding the Function

The given function is \( f(x) = x^2 e^{-2x} \). This can be rewritten using the exponential rules as \( f(x) = \frac{x^2}{e^{2x}} \).

Step 2: Finding \( \lim_{x \rightarrow +\infty} f(x) \)

We rewrite \( f(x) \) as \( \frac{x^2}{e^{2x}} \). As \( x \to +\infty \), the numerator \( x^2 \to +\infty \) and denominator \( e^{2x} \to +\infty \) as well. By L'Hôpital's Rule, since both the numerator and the denominator tend towards infinity, evaluate:\[\lim_{x \to +\infty} \frac{2x}{2e^{2x}} = \lim_{x \to +\infty} \frac{x}{e^{2x}}\]Using the known result, \( \lim_{x \rightarrow +\infty} \frac{x}{e^{x}} = 0 \), it follows that:\[\lim_{x \rightarrow +\infty} \frac{x}{e^{2x}} = 0,\]thus \( \lim_{x \right ] \rightarrow +\infty} f(x) = 0 \).

Step 3: Finding \( \lim_{x \rightarrow -\infty} f(x) \)

Next, consider \( \lim_{x \rightarrow -\infty} x^2 e^{-2x} \). As \( x \to -\infty \), \( x^2 \to +\infty \) and \( e^{-2x} \to +\infty \) because it behaves as \( e^{+2|x|} \). Hence, \( f(x) \) grows towards infinity, therefore:\[\lim_{x \rightarrow -\infty} f(x) = +\infty.\]

Step 4: Identifying Behavior and Graph Features

Now, to sketch the graph and identify features:- Asymptotes: Since \( \lim_{x \rightarrow +\infty} f(x) = 0 \), there is a horizontal asymptote at \( y = 0 \) for large \( x \).- As \( x \rightarrow -\infty \), \( f(x) \to \infty \).- **To find critical points**, take derivative \( f'(x) \) and set \( f'(x) = 0 \) to locate extrema.- **Inflection points** can be determined by finding where the second derivative changes sign.Use a graphing utility to confirm these points.

Key Concepts

Limits at Infinity

When we explore functions that approach certain values as the input grows indefinitely large (either positively or negatively), we are dealing with limits at infinity. For example, consider the function

• For positive infinity: As the input \( x \) moves toward positive infinity, the squared term \( x^2 \) within the function \( f(x) = \frac{x^2}{e^{2x}} \) grows very large, but the denominator \( e^{2x} \) grows even faster, ultimately driving the fraction towards zero.
• For negative infinity: Here, the exponential term \( e^{-2x} \) technically becomes \( e^{+2|x|} \), also leading both numerator and denominator to infinity. However, the negative exponent \( -2x \) makes the exponential function significant, pushing the value of \( f(x) \) to infinity.

These behaviors illustrate the overarching principles at play — namely, how exponential growth can outpace polynomial growth. Using L'Hôpital's Rule, a method applied when forms like \( \frac{\infty}{\infty} \) or \( \frac{0}{0} \) are encountered, helps confirm these limits by differentiating the respective components of the fraction.

Exponential Functions

Exponential functions are powerful mathematical expressions often characterized by their rapid growth or decay, depending on their exponent. The general form of an exponential function is \( e^x \), where \( e \) denotes Euler's number, approximately equal to 2.718.

In the function \( f(x) = x^2 e^{-2x} \) can be outlined as follows:

• When exponential terms include negative exponents, forces within the function decrease rapidly as \( x \) increases.
• The negative exponent \(-2x\) specifically means the function decreases rapidly, behaving oppositely compared to when it has a positive exponent.

The inclusion of such a term can completely alter the function's behavior, as seen when \( f(x) \) approaches the limits. Regardless of the polynomial factor \( x^2 \), the power of the exponential function ultimately dictates the limit properties and overall characteristic of \( f(x) \). This is notably crucial when evaluating the behavior as \( x \) approaches infinity.

Sketching Graphs

To successfully sketch a graph of a function like \( f(x) = x^2 e^{-2x} \), several steps and considerations must be undertaken:

• **Limits and Asymptotes**: Recognize the horizontal asymptote identified at \( y = 0 \) as \( x \) tends to positive infinity. This information assists in anticipating the graph's trajectory.
• **Critical Points**: Calculate the derivative \( f'(x) \) and set it to zero to discover local maxima or minima, often referred to as relative extrema, providing essential turning points of the graph.
• **Inflection Points**: By analyzing where the second derivative \( f''(x) \) changes signs, one can pinpoint where the graph's curvature transitions from concave down to concave up, or vice versa.

Utilizing a graphing utility aids in visually reaffirming these attributes, allowing a deeper understanding of the function's behavior visually and mathematically. This synthesis of calculus principles is imperative in constructing accurate and informative graphical representations.