BackReal Analysis Assignment: Limits and Differentiability of a Cubic Function
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Q1. Determine for .
Background
Topic: Limits at Infinity for Polynomial Functions
This question tests your understanding of how to analyze the behavior of a polynomial function as approaches infinity, a fundamental concept in calculus and real analysis.
Key Terms and Formulas:
Limit at Infinity: The value that a function approaches as becomes arbitrarily large.
Polynomial Growth: For polynomials, the highest degree term dominates as .
Step-by-Step Guidance
Identify the degree of each term in . The function $f(x)$ is a cubic polynomial, so the term will dominate as increases.
Rewrite to highlight the dominant term: .
As , compare the growth rates of each term. The term grows much faster than the linear () and constant () terms.
To analyze the limit, factor from each term to see how the other terms behave relative to $x^3$ as becomes very large.
Try solving on your own before revealing the answer!
Final Answer:
As , the cubic term dominates, so increases without bound.
The linear and constant terms become negligible compared to the cubic term, so the limit is .
