Limits: Quick Review

Key Concepts

Direct substitution is valid for continuous functions. If f is continuous at a, then $\lim<em>{x\to a} f(x) = f(a)$. Polynomials are continuous everywhere, so limits equal the evaluation: $\lim</em>{x\to a} p(x) = p(a)$.
When the limit yields a form 0/0, algebraic manipulation (factoring, canceling common factors) is used to simplify before substitution.
When the limit involves infinity, analyze growth rates: if the dominant term grows without bound, the limit is ±∞.

Direct Substitution and Simple Finite Limits

If the function is simple and continuous, substitute the value. Examples: $\lim<em>{x\to 3} x^2 = 9$, $\lim</em>{x\to 1} (2x+1) = 3$, $\lim_{x\to 2} 2x = 4$.

Finite Polynomial Limits from the Transcript

From the transcript, certain explicit evaluations are correct:

• $\lim_{x\to 0} x^2 = 0$.
• $\lim_{x\to 3} x^2 = 9$.
• $\lim_{x\to 2} (2x^2 - 2x + 1) = 5$.

Limits at Infinity

For expressions that grow without bound as x grows, the limit is infinite: $\lim_{x\to \infty} (2x^2 + 1) = \infty$.

Quick Reference Card

• If f is continuous at a: $\lim_{x\to a} f(x) = f(a)$.
• For polynomials: $\lim_{x\to a} p(x) = p(a)$.
• For simple linear: $\lim_{x\to a} (mx + b) = ma + b$.
• For rational expressions where the denominator tends to 0, check for factorization to cancel the zero if possible; otherwise, the limit may diverge.

Notes on the Transcript

Some parts of the provided transcript are garbled; the notes above capture the reliable, commonly applicable limit results and the explicit correct evaluations that were clear in the text.