Problem 24 - Calculus: Early Transcendentals

Question

Determine the following limits. $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right)$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches negative infinity is $$-\infty$$.

Step 1: Determine the dominant term

As the limit approaches infinity, the values of the exponential terms will control the values of the function. To compare the two terms, we focus on the powers of x. The powers of x in the polynomial are -8 and 3. Since a higher positive power grows faster, the term with power 3 will dominate as x approaches negative infinity. Step 2:

Step 2: Evaluate the limit

Since the dominant term is the one with power 3, we consider only this term when calculating the limit: $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right) = \lim _{x \rightarrow-\infty}\left(4 x^{3}\right)$$ Step 3:

Step 3: Apply the limit

Since we are now evaluating the limit for a single term, we can directly substitute the value of x as negative infinity: $$\lim _{x \rightarrow-\infty}\left(4 x^{3}\right) = 4 (-\infty )^{3} = -\infty$$ Step 4:

Step 4: Final answer

The final answer is, therefore, $$\lim _{x \rightarrow-\infty}\left(2 x^{-8}+4 x^{3}\right) = -\infty$$

Key Concepts

Dominant Term

When evaluating the limits of a polynomial function as it approaches infinity, identifying the dominant term is crucial. In a polynomial, each term is composed of a coefficient and a power of the variable, commonly known as "x". The dominant term is the one with the highest power of x.

This term dictates the behavior of the entire polynomial as it stretches towards positive or negative infinity. In our exercise, we have two terms: one with a power of -8 and another with power 3.

• Powers lower than the dominant contribute minimally to the overall value.
• The dominant term's rate of growth surpasses others as x moves towards infinity.
• To identify the dominant term, compare power sizes; here, 4x³ is dominant because 3 > -8.

Understanding the dominant term allows us to simplify complex polynomial expressions, making it easier to calculate limits.

Polynomial Limits

Polynomial limits are often determined while a variable approaches infinity, positive or negative. With polynomials, the approach often involves recognizing the most significant term affecting the limit as x approaches either infinity or negative infinity.

• First, identify the dominant term, as it impacts the limit most significantly at extreme values.
• Recognize that lower power terms diminish in significance as x goes towards large values.
• The limit value often mirrors the nature of the dominant term; if it tends towards negative or positive infinity, so does the limit.

In this exercise, the term 4x³ dominates, leading the polynomial limit, thus simplifying the calculation to evaluate:\[\lim _{x \rightarrow -\infty} 4x^{3}.\]This offers insight into how powerful terms drive the polynomial's overall behavior in determining limits.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input grows larger and larger, heading towards infinity or negative infinity. Recognizing how a function behaves asymptotically means understanding which terms affect its growth and how these terms contribute to the function's limit at far ends of the graph.

Particularly in polynomials:

• The leading term dictates the asymptotic direction, giving insight into whether the function grows positively or negatively.
• In our exercise, the term 4x³ not only leads but indicates that as x shrinks to negative infinity, the function tends toward negative infinity, indicating a steep descent.
• Asymptotic behavior showcases the dominant term's influence—understanding this leads to predictions about the graph's shape and direction.

Observing asymptotic behavior through dominant terms helps in anticipating how functions behave in real-world scenarios over large scales.