Limits and Continuity

Limits and Continuity

Limits describe the behavior of a function as its input approaches a particular value. Here's what you need to know:

Basic Concept of Limits

Definition

The notation $$\lim_{x \to a} f(x) = L$$ means that as x gets arbitrarily close to a (but not equal to a), the value of f(x) gets arbitrarily close to L.

One-Sided Limits

• Left-hand limit: $$\lim_{x \to a^-} f(x)$$ (approaching from values less than a)
• Right-hand limit: $$\lim_{x \to a^+} f(x)$$ (approaching from values greater than a)

A limit exists if and only if both one-sided limits exist and are equal:

$$\lim_{x \to a} f(x) = L$$ if and only if $$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$

Infinite Limits

If f(x) grows without bound as x approaches a, we write:

$$\lim_{x \to a} f(x) = \infty$$ or $$\lim_{x \to a} f(x) = -\infty$$

Limits at Infinity

Describes the behavior of f(x) as x becomes arbitrarily large:

$$\lim_{x \to \infty} f(x) = L$$

Similarly, we can examine the behavior as x approaches negative infinity:

$$\lim_{x \to -\infty} f(x) = L$$

Techniques for Evaluating Limits

Direct Substitution

If f is continuous at a, then $$\lim_{x \to a} f(x) = f(a)$$

Example: $$\lim_{x \to 2} (x^2 + 3x) = 2^2 + 3(2) = 10$$

Algebraic Manipulation

For limits of the form $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ where f(a) = g(a) = 0:

1. Factor the numerator and denominator
2. Cancel common factors
3. Substitute x = a

Example: $$\lim_{x \to 3} \frac{x^2-9}{x-3} = \lim_{x \to 3} \frac{(x-3)(x+3)}{x-3} = \lim_{x \to 3} (x+3) = 6$$

Limits of Rational Functions as x → ∞

For $$\lim_{x \to \infty} \frac{a_n x^n + ... + a_1 x + a_0}{b_m x^m + ... + b_1 x + b_0}$$:

• If n < m: limit = 0
• If n = m: limit = $$\frac{a_n}{b_m}$$
• If n > m: limit = ∞ or -∞ (depending on signs of leading coefficients)

L'Hôpital's Rule

If $$\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$$ or $$\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = \pm\infty$$, then:

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$

provided the limit on the right exists or is infinite.

Continuity

Definition

A function f is continuous at a point a if:

1. f(a) is defined
2. $$\lim_{x \to a} f(x)$$ exists
3. $$\lim_{x \to a} f(x) = f(a)$$

Types of Discontinuities

• Removable discontinuity: The limit exists but doesn't equal f(a) or f(a) is undefined
• Jump discontinuity: Left and right limits exist but are not equal
• Infinite discontinuity: The limit is infinite
• Oscillating discontinuity: The limit doesn't exist due to oscillation

Properties of Continuous Functions

• Intermediate Value Theorem: If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in [a,b] such that f(c) = k
• Extreme Value Theorem: If f is continuous on a closed interval [a,b], then f attains both a maximum and minimum value on [a,b]

Important Limits

Fundamental Limits

• $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$
• $$\lim_{x \to 0} \frac{1-\cos x}{x} = 0$$
• $$\lim_{x \to 0} (1+x)^{1/x} = e$$
• $$\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$$

Common Pitfalls and Tips

• Remember that $$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$$ only if both individual limits exist
• Be careful with expressions like $$\frac{0}{0}$$ or $$\frac{\infty}{\infty}$$ (indeterminate forms)
• When using L'Hôpital's Rule, make sure you're dealing with an appropriate indeterminate form
• For piecewise functions, check continuity at the transition points
• When finding limits at infinity, identify the dominant terms