IB Mathematics AA HL Flashcards -Informal ideas of limit, continuity and convergence

CONTINUITY ⟶ A function can be continuous at a point, continuous on an interval, or simply just continuous. To be continuous at a point, f must be defined at ‘a’, the limit of f at ‘a’ exists, and f(a) = \(\lim_{{x \to a}} f(x)\). But this essentially means there is an uninterrupted line, with no holes, abrupt breaks, or jumps. If it is continuous at all points in the domain, f is a continuous function.

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DIFFERENTIABILITY ⟶ If f'(x₁) exists, then f is differentiable at x₁. If f is differentiable at every point in its domain, the function itself is differentiable.

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LIMITS ⟶ They were discussed in 5.1 and will be discussed further later on as well. Here, we will see one simple type of limit evaluation for rational functions, for x → ∞. First, we must note the following trivial facts:

\[
\lim_{{x \to ∞}} (a x) = ∞ \quad \cdot \quad \lim_{{x \to ∞}} (c) = c \quad \cdot \quad \lim_{{x \to ∞}} \left(\frac{a}{x}\right) = 0
\]

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Now, take the following three solutions:

1. \[
\lim_{{x \to ∞}} \left(\frac{2x^2 + 5x}{7x – 3x^2}\right) = \lim_{{x \to ∞}} x \cdot \lim_{{x \to ∞}} \left(\frac{2x^2 + 5}{7 – 3x^2}\right) = ∞ \cdot \frac{2}{3} = ∞ \quad \text{→ it ‘diverges’}
\]

2. \[
\lim_{{x \to ∞}} \left(\frac{4x^3 + 6x}{7x – 2x^2}\right) \quad \text{divide by x²} \quad \lim_{{x \to ∞}} \left(\frac{4x + 6}{7 – 2x}\right) = \lim_{{x \to ∞}} \left(\frac{4 + 0}{0 – 7}\right) = \frac{4}{-7} \quad \text{→ it ‘converges’}
\]

3. \[
\lim_{{x \to ∞}} \left(\frac{x + 8}{3x^2 + 11}\right) \quad \text{divide by x²} \quad \lim_{{x \to ∞}} \left(\frac{\frac{x}{x^2} + \frac{8}{x^2}}{3 + \frac{11}{x^2}}\right) = \lim_{{x \to ∞}} \left(\frac{0 + 0}{3 + 0}\right) = 0 \quad \text{→ it ‘converges’}
\]

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⟶ These sum up the three conditions of the following rule:

If you have \(\lim_{{x \to ∞}} \left(\frac{P(x)}{Q(x)}\right)\), where P & Q are polynomials, then:

\[
\lim_{{x \to ∞}} = ∞ \quad \text{if degree of P > degree of Q}
\]
\[
\lim_{{x \to ∞}} = 0 \quad \text{if degree of P < degree of Q}
\]
\[
\lim_{{x \to ∞}} = \frac{a}{b} \quad \text{if degrees are both n, and a & b are coefficients of } x^n \text{ in P & Q.}
\]

DERIVATIVES FROM 1st PRINCIPLES ⟶ If you remember from 5.1, the actual definition of a derivative involves limits: \(f'(x) = \lim_{{h \to 0}} \frac{f(x + h) – f(x)}{h}\)

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