Understanding limits is fundamental to mastering calculus, particularly in the study of continuity and differentiation. In the International Baccalaureate (IB) Mathematics: AI SL curriculum, limits serve as the foundation for exploring more complex mathematical concepts. This article delves into the definition and calculation of limits, providing a comprehensive guide tailored for IB students.

A limit describes the value that a function approaches as the input approaches a particular point. Formally, the limit of a function \( f(x) \) as \( x \) approaches \( c \) is denoted as: $$ \lim_{x \to c} f(x) = L $$ This means that as \( x \) gets closer to \( c \), \( f(x) \) gets arbitrarily close to \( L \). Limits are essential in defining continuous functions and derivatives.

Limits can be approached from the left or the right, leading to the concepts of left-hand limits and right-hand limits. - **Left-Hand Limit**: $$ \lim_{x \to c^-} f(x) = L $$ This represents the value \( f(x) \) approaches as \( x \) approaches \( c \) from values less than \( c \). - **Right-Hand Limit**: $$ \lim_{x \to c^+} f(x) = L $$ This denotes the value \( f(x) \) approaches as \( x \) approaches \( c \) from values greater than \( c \). For the overall limit \( \lim_{x \to c} f(x) \) to exist, both one-sided limits must exist and be equal.

Several methods exist for calculating limits, each applicable depending on the function's behavior and the point of interest. 3.1. Direct Substitution For continuous functions at \( x = c \), the limit can often be found by directly substituting \( c \) into \( f(x) \): $$ \lim_{x \to c} f(x) = f(c) $$ *Example:* Find \( \lim_{x \to 2} (3x + 4) \). $$ \lim_{x \to 2} (3x + 4) = 3(2) + 4 = 10 $$ 3.2. Factoring When direct substitution results in an indeterminate form like \( \frac{0}{0} \), factoring can simplify the expression. *Example:* Find \( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \). $$ \frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3} = x + 3 \quad \text{for } x \neq 3 $$ Thus, $$ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} = 3 + 3 = 6 $$ 3.3. Rationalizing For limits involving radicals, multiplying by the conjugate can eliminate the radical. *Example:* Find \( \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} \). $$ \frac{\sqrt{x} - 2}{x - 4} \times \frac{\sqrt{x} + 2}{\sqrt{x} + 2} = \frac{x - 4}{(x - 4)(\sqrt{x} + 2)} = \frac{1}{\sqrt{x} + 2} \quad \text{for } x \neq 4 $$ Thus, $$ \lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4} = \frac{1}{2 + 2} = \frac{1}{4} $$ 3.4. Using L'Hôpital's Rule When direct substitution yields \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule can be applied: $$ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \quad \text{if } \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0 \text{ or } \pm\infty $$ *Example:* Find \( \lim_{x \to 0} \frac{\sin x}{x} \). Applying L'Hôpital's Rule: $$ \lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1 $$

Limits can also describe the behavior of functions as \( x \) approaches infinity or negative infinity. 4.1. Horizontal Asymptotes Determining \( \lim_{x \to \infty} f(x) \) helps identify horizontal asymptotes. *Example:* Find \( \lim_{x \to \infty} \frac{2x + 3}{x - 1} \). Divide numerator and denominator by \( x \): $$ \lim_{x \to \infty} \frac{2 + \frac{3}{x}}{1 - \frac{1}{x}} = \frac{2 + 0}{1 - 0} = 2 $$ Thus, \( y = 2 \) is a horizontal asymptote. 4.2. Limits at Infinity for Polynomials For a polynomial \( f(x) = a_nx^n + \dots + a_0 \), the limit as \( x \) approaches infinity is determined by the leading term: $$ \lim_{x \to \infty} f(x) = \begin{cases} \infty & \text{if } a_n > 0 \text{ and } n \text{ is even} \\ -\infty & \text{if } a_n < 0 \text{ and } n \text{ is even} \\ \end{cases} $$ *Example:* Find \( \lim_{x \to \infty} (5x^3 - 2x + 7) \). Since the leading term is \( 5x^3 \) and its coefficient is positive: $$ \lim_{x \to \infty} (5x^3 - 2x + 7) = \infty $$

A function is continuous at \( x = c \) if: $$ \lim_{x \to c} f(x) = f(c) $$ Understanding limits is crucial for determining continuity. If either one-sided limits do not exist or are not equal to \( f(c) \), the function is discontinuous at that point.

The Squeeze Theorem is useful for finding limits of functions trapped between two other functions. If: $$ g(x) \leq f(x) \leq h(x) \quad \text{for all } x \text{ near } c $$ and $$ \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L $$ then, $$ \lim_{x \to c} f(x) = L $$ *Example:* Find \( \lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) \). Since \( -x^2 \leq x^2 \cos\left(\frac{1}{x}\right) \leq x^2 \) and both \( \lim_{x \to 0} -x^2 = 0 \) and \( \lim_{x \to 0} x^2 = 0 \), $$ \lim_{x \to 0} x^2 \cos\left(\frac{1}{x}\right) = 0 $$

Understanding the properties of limits allows for the simplification and calculation of complex limit expressions. 7.1. Sum/Difference Rule $$ \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x) $$ 7.2. Constant Multiple Rule $$ \lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x) $$ 7.3. Product Rule $$ \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) $$ 7.4. Quotient Rule $$ \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \quad \text{if } \lim_{x \to c} g(x) \neq 0 $$> 7.5. Power and Root Rules $$ \lim_{x \to c} [f(x)]^n = \left( \lim_{x \to c} f(x) \right)^n $$ $$ \lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)} $$>

When evaluating limits, certain forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) are considered indeterminate because they do not directly reveal the limit's value. Techniques such as factoring, rationalizing, or L'Hôpital's Rule are employed to resolve these forms.

Example 1: Evaluate \( \lim_{x \to 1} \frac{x^2 - 1}{x - 1} \). \begin{align*} \frac{x^2 - 1}{x - 1} &= \frac{(x - 1)(x + 1)}{x - 1} \\ &= x + 1 \quad \text{for } x \neq 1 \\ \lim_{x \to 1} (x + 1) &= 2 \end{align*} Example 2: Find \( \lim_{x \to 0} \frac{\sin x}{x} \). Using L'Hôpital's Rule: \begin{align*} \lim_{x \to 0} \frac{\sin x}{x} &= \lim_{x \to 0} \frac{\cos x}{1} \\ &= \cos 0 \\ &= 1 \end{align*} Example 3: Determine \( \lim_{x \to \infty} \frac{3x^2 + 2x + 1}{6x^2 - x + 4} \). Divide numerator and denominator by \( x^2 \): \begin{align*} \lim_{x \to \infty} \frac{3 + \frac{2}{x} + \frac{1}{x^2}}{6 - \frac{1}{x} + \frac{4}{x^2}} &= \frac{3 + 0 + 0}{6 - 0 + 0} \\ &= \frac{3}{6} \\ &= \frac{1}{2} \end{align*}

• Limits describe the behavior of functions as inputs approach specific values.
• One-sided limits provide insights from the left and right perspectives.
• Various techniques, including factoring and L'Hôpital's Rule, are essential for calculating limits.
• Understanding limits is crucial for studying continuity and derivatives in calculus.
• Proper application of limit properties simplifies the evaluation of complex expressions.