limits

1. Definition of Limits

• A limit is the value that a function approaches as the input approaches some value.

• Notation: $\lim_{x \to c} f(x) = L$ means that as $x$ approaches $c$, $f(x)$ approaches $L$.

2. Understanding Limits Graphically

• Limits can often be evaluated by observing the graph of the function.

• Check the behavior of the function from both the left-hand limit and the right-hand limit.

3. Evaluating Limits Algebraically

• Direct Substitution: If $f(c)$ is defined, then $\lim_{x \to c} f(x) = f(c)$.

• Factoring: Factor the expression and simplify before evaluating.

• Rationalizing: Multiply by the conjugate to eliminate radicals in the numerator or denominator.

4. Special Limits

• Limit at Infinity: $\lim_{x \to \infty} f(x)$ considers the value of the function as $x$ becomes very large.

• One-Sided Limits:

  • Left-hand limit: $\lim_{x \to c^-} f(x)$

  • Right-hand limit: $\lim_{x \to c^+} f(x)$

5. The Squeeze Theorem

• If $g(x) \leq f(x) \leq h(x)$ for all $x$ in some neighborhood of $c$ (excluding possibly at $c$ itself), and if $\lim<em>{x \to c} g(x) = \lim</em>{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$.

6. Infinite Limits and Limits at Infinity

• An infinite limit means the function increases or decreases without bound as it approaches a certain point.

• Notation: $\lim<em>{x \to c} f(x) = \infty$ or $\lim</em>{x \to c} f(x) = -\infty$.

• For limits at infinity, rational functions have particular behaviors based on degrees of the numerator and denominator.

7. Common Limit Rules

• Sum Rule: $\lim<em>{x \to c} [f(x) + g(x)] = \lim</em>{x \to c} f(x) + \lim_{x \to c} g(x)$

• Product Rule: $\lim<em>{x \to c} [f(x) \cdot g(x)] = \lim</em>{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

• Quotient Rule: $\lim<em>{x \to c} \frac{f(x)}{g(x)} = \frac{\lim</em>{x \to c} f(x)}{\lim<em>{x \to c} g(x)}$ (provided $\lim</em>{x \to c} g(x) \neq 0$)

8. Practice Problems

1. Evaluate $\lim_{x \to 3} (2x + 1)$

2. Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

3. Evaluate $\lim_{x \to \infty} \frac{5x^3 + 2}{2x^3 - 4}$