Key Concepts of Limits of Functions

Limits of functions help us understand how a function behaves as it gets close to a certain point. This concept is crucial in Precalculus, laying the groundwork for calculus by exploring continuity, one-sided limits, and the behavior of functions at infinity.

1. Definition of a limit

   • A limit describes the value that a function approaches as the input approaches a certain point.
   • Limits can be finite or infinite, depending on the behavior of the function.
   • The notation ( \lim_{x \to c} f(x) = L ) indicates that as ( x ) approaches ( c ), ( f(x) ) approaches ( L ).

2. One-sided limits

   • One-sided limits consider the behavior of a function as it approaches a point from one side: the left or the right.
   • The left-hand limit is denoted as ( \lim_{x \to c^-} f(x) ) and the right-hand limit as ( \lim_{x \to c^+} f(x) ).
   • For a limit to exist at a point, both one-sided limits must be equal.

3. Limit laws (sum, difference, product, quotient)

   • Limit laws provide rules for calculating limits of combined functions.
   • The sum law states ( \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) ).
   • The product law states ( \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) ).
   • The quotient law states ( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} ) (provided ( g(c) \neq 0 )).

4. Limits involving infinity

   • Limits can also describe the behavior of functions as the input approaches infinity or negative infinity.
   • The notation ( \lim_{x \to \infty} f(x) ) indicates the value that ( f(x) ) approaches as ( x ) increases without bound.
   • If a function approaches a finite value as ( x ) approaches infinity, it is said to have a horizontal asymptote.

5. Limits at infinity

   • Limits at infinity focus on the end behavior of functions as ( x ) approaches positive or negative infinity.
   • Polynomial functions can be analyzed to determine their end behavior based on the leading term.
   • Rational functions may have horizontal asymptotes determined by the degrees of the numerator and denominator.

6. Continuity and limits

   • A function is continuous at a point if the limit at that point equals the function's value.
   • For continuity, three conditions must be met: the function must be defined at the point, the limit must exist, and the limit must equal the function's value.
   • Discontinuities can be classified as removable, jump, or infinite.

7. Evaluating limits algebraically

   • Algebraic techniques such as substitution, factoring, and rationalizing can be used to evaluate limits.
   • If direct substitution results in an indeterminate form (like ( \frac{0}{0} )), further manipulation is needed.
   • Common techniques include simplifying expressions and applying limit laws.

8. Evaluating limits graphically

   • Graphical evaluation involves analyzing the graph of a function to determine its behavior near a specific point.
   • Observing the y-values as x approaches a certain value can provide insight into the limit.
   • Discontinuities and asymptotic behavior can often be identified visually.

9. Limits of trigonometric functions

   • Trigonometric limits often involve special angles and can be evaluated using known limit values.
   • Key limits include ( \lim_{x \to 0} \frac{\sin x}{x} = 1 ) and ( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} ).
   • Understanding the periodic nature of trigonometric functions is essential for evaluating limits.

10. Squeeze theorem

    • The Squeeze theorem is used to find limits of functions that are "squeezed" between two other functions with known limits.
    • If ( g(x) \leq f(x) \leq h(x) ) and ( \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L ), then ( \lim_{x \to c} f(x) = L ).
    • This theorem is particularly useful for evaluating limits of functions that oscillate or are difficult to analyze directly.