Problem 291 - Calculus Volume 1

Question

For the following exercises, graph the function on a graphing calculator on the window \(x=[-5,5]\) and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. $$ \lim _{x \rightarrow-\infty} x^{2}+10 x+25 $$

Step-by-Step Solution

Answer

The function does not have a horizontal asymptote as \( x \to -\infty \).

Step 1: Recognize the Limit at Infinity

The expression we are examining is \( \lim_{x \to -\infty} x^2 + 10x + 25 \). To find the horizontal asymptote or limit, observe the behavior of the polynomial as \( x \to -\infty \).

Step 2: Analyze Dominant Term

In the polynomial \( x^2 + 10x + 25 \), the term \( x^2 \) grows faster than the other terms as \( x \to -\infty \). Therefore, \( x^2 \) is the dominant term determining the behavior of the function.

Step 3: Apply Limit

Taking the limit as \( x \to -\infty \) for the polynomial \( x^2 + 10x + 25 \), we note that the term \( x^2 \) will cause the overall value of the expression to approach \( \infty \).

Step 4: Conclude Horizontal Asymptote

Since the leading term \( x^2 \) dominates and grows indefinitely, there is no horizontal asymptote as \( x \to -\infty \). The function does not approach a constant value.

Key Concepts

Horizontal Asymptote

A horizontal asymptote represents a horizontal line that a graph approaches as the independent variable. For our case, the variable is \( x \) and it goes towards positive or negative infinity. These lines show us the behavior of functions at extreme values and tell us if a function is stabilizing towards a particular constant value.
To determine a horizontal asymptote, particularly for rational functions, we compare the degrees of the polynomial in the numerator and the denominator. For example:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, the horizontal asymptote is \( y = \frac{a}{b} \), where \( a \) and \( b \) are the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, no horizontal asymptote exists.

However, in our problem, only a polynomial is present. This means there aren't separate degrees to compare. Hence, as seen through analyzing it, this quadratic polynomial, \( x^2 + 10x + 25 \), approaches infinity as \( x \to -\infty \), so there is no horizontal asymptote.

Dominant Term

In polynomial expressions, not all terms affect the behavior of the function equally as \( x \) becomes very large or very small. In such cases, the dominant term essentially "takes over", dictating the overall behavior.
The dominant term is the one with the highest degree, and it grows faster than any other terms. For example, in \( x^2 + 10x + 25 \), \( x^2 \) is the dominant term because it's the highest degree term.
This dominant term will typically determine the direction the polynomial heads. While smaller degree terms like \( 10x \) or constants like \( 25 \) might seem important, they diminish in influence as \( x \) moves towards the extremes of infinity. Recognizing the dominant term simplifies analysis of polynomials at infinity.

Polynomial Behavior

Polynomials display predictable patterns as \( x \) approaches either positive or negative infinity, dictated by their dominant terms. Understanding the behavior involves observing the sign and degree of this dominant term.
For instance:

• If the degree is even and leading coefficient positive, the function rises on both ends.
• If it is even and the coefficient negative, it falls on both ends.
• For odd degrees with a positive coefficient, the graph will fall on the left and rise on the right.
• With an odd degree and a negative coefficient, the function rises on the left and falls on the right.

In our problem, \( x^2 + 10x + 25 \), the degree is 2 (even) and the leading coefficient is 1 (positive). This means, the polynomial rises indefinitely as \( x \to -\infty \), further confirming there are no horizontal asymptotes. The predictable nature of polynomials and recognizing dominant terms make it easier to anticipate their trends and interpret limits at infinity.