Problem 30 - Calculus

Question

For each function, fill in the blanks in the statements: \(f(x) \rightarrow \quad\) as \(x \rightarrow-\infty\) \(f(x) \rightarrow \quad\) as \(x \rightarrow+\infty\) (a) \(f(x)=17+5 x^{2}-12 x^{3}-5 x^{4}\) (b) \(f(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}-8}\) (c) \(f(x)=e^{x}\)

Step-by-Step Solution

Answer

(a) Both limits: \(-\infty\). (b) Both limits: \(\frac{3}{2}\). (c) As \(x\rightarrow -\infty\), \(0\); as \(x\rightarrow +\infty\), \(+\infty\).

Step 1: Identify the Leading Term (a)

For the polynomial function \(f(x)=17+5x^2-12x^3-5x^4\), the leading term is the one with the highest power of \(x\). Here, it is \(-5x^4\). As \(x\) approaches \(-\infty\) and \(+\infty\), the sign of the leading term will determine the behavior of \(f(x)\).

Step 2: Analyze the Behavior with the Leading Term (a)

For large |x|, the term \(-5x^4\) dominates the polynomial. - As \(x\rightarrow -\infty\), \(-5x^4\rightarrow -\infty\) because raising a negative number to an even power becomes positive, and multiplying by a negative coefficient makes it negative.- As \(x\rightarrow +\infty\), \(-5x^4\rightarrow -\infty\) because the term remains negative.

Step 3: Fill in the Blanks for Part (a)

Using the analysis from Step 2: \(f(x)\rightarrow -\infty\) as \(x\rightarrow -\infty\)\(f(x)\rightarrow -\infty\) as \(x\rightarrow +\infty\)

Step 4: Simplify the Rational Function (b)

For \(f(x)=\frac{3x^2-5x+2}{2x^2-8}\), express \(f(x)\) as:\[f(x) = \frac{3x^2}{2x^2} - \frac{5x}{2x^2} + \frac{2}{2x^2} \approx \frac{3}{2} \text{ as } x \rightarrow \pm\infty\]So, the horizontal asymptote is \(y=\frac{3}{2}\).

Step 5: Find Limits for Part (b)

As \(x \rightarrow \pm\infty\), the function approaches its horizontal asymptote.\(f(x)\rightarrow \frac{3}{2}\) as \(x \rightarrow -\infty\) \(f(x)\rightarrow \frac{3}{2}\) as \(x \rightarrow +\infty\)

Step 6: Analyze the Exponential Function (c)

For \(f(x)=e^x\):- As \(x\rightarrow -\infty\), \(e^x\rightarrow 0\) because the exponent is a large negative number making \(e^x\) approach zero.- As \(x\rightarrow +\infty\), \(e^x\rightarrow +\infty\) because the exponent is a large positive number.

Step 7: Fill in the Blanks for Part (c)

From the analysis in Step 6:\(f(x)\rightarrow 0\) as \(x\rightarrow -\infty\)\(f(x)\rightarrow +\infty\) as \(x\rightarrow +\infty\)

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions composed of variables and coefficients, bound together by addition, subtraction, or multiplication operations. They often take the form of terms like \(ax^n\), where \(n\) is a non-negative integer. The largest power of \(x\) in the polynomial determines the degree of the polynomial, and the term with this power is referred to as the leading term. For example, in the polynomial \(f(x) = 17 + 5x^2 - 12x^3 - 5x^4\), the term \(-5x^4\) is the leading term. This is crucial because the behavior of polynomial functions at the extremes (as \(x\) approaches \(-\infty\) or \(+\infty\)) is largely dictated by the leading term.

When analyzing the asymptotic behavior of polynomial functions:

• For an even-degree leading term with a negative coefficient, like \(-5x^4\), the function moves towards \(-\infty\) as \(x\) approaches both \(-\infty\) and \(+\infty\).
• The opposite applies if the coefficient is positive; the function would head towards \(+\infty\) on both ends.

Rational Functions

Rational functions are expressed as the ratio of two polynomial functions. They resemble fractions, where both the numerator and denominator are polynomials. For example, in the function \(f(x) = \frac{3x^2 - 5x + 2}{2x^2 - 8}\), the numerator is \(3x^2 - 5x + 2\), and the denominator is \(2x^2 - 8\). A key aspect of understanding rational functions is identifying their asymptotic behavior, especially the horizontal asymptotes. This occurs when the degrees of the numerator and denominator are equal.

To find horizontal asymptotes:

• Compare the degrees of the polynomials in the numerator and the denominator. If they are equal, the horizontal asymptote is found by dividing the leading coefficients, here \(\frac{3}{2}\).
• If the degree of the numerator is less, the horizontal asymptote is \(y = 0\).
• If the degree of the numerator is greater, there is no horizontal asymptote.

Exponential Functions

Exponential functions feature a constant base raised to a power that is a variable, typically expressed as \(f(x) = a^x\), where \(a\) is a positive constant. In particular, when the base is the natural constant \(e\) (approximately 2.718), the function becomes \(f(x) = e^x\). These functions exhibit unique behavior not seen in polynomial or rational functions.

Key characteristics of exponential functions include:

• As \(x\) approaches \(-\infty\), \(e^x\) decreases towards zero, due to the negative exponent reducing the function's value.
• Conversely, as \(x\) approaches \(+\infty\), \(e^x\) increases rapidly, effectively growing without bound towards \(+\infty\).

The distinctiveness of exponential functions lies in their growth speed and behavior in differing extreme values of \(x\). This emerging pattern is a crucial concept in calculus and broader mathematical applications.