5 Limits of Functions
In the previous chapter Operations on Functions, we explored how to combine functions through addition, subtraction, multiplication, division, and composition. These operations allowed us to model multi-step processes and represent interactions between different mathematical and real-world quantities.
In this chapter, we extend that understanding by studying Limits of Functions, which analyze the behavior of functions as their inputs approach specific points or infinity. Limits are a fundamental concept in calculus, forming the foundation for continuity, derivatives, and integrals. They allow us to rigorously describe how functions behave near critical points, handle discontinuities, and model phenomena where changes are instantaneous or unbounded.
This Figure 5.1 will cover:
- The definition of limits and their interpretation
- Types of limits, including finite, infinite, one-sided, and limits at infinity
- Techniques for evaluating limits, such as substitution, factoring, rationalization, and L’Hospital’s Rule
- Special limits involving trigonometric, exponential, and logarithmic functions
- Real-world applications, such as instantaneous velocity, marginal cost, population growth, and rates of change in engineering and natural sciences
By the end of this chapter, you will be able to evaluate limits, understand the behavior of functions near points of interest, and apply these concepts to model and analyze real-world systems.
5.1 Review
Before diving into the formal definitions, let’s use an analogy to understand what a limit is: Imagine you are driving a car toward a stop sign. As you get closer, your speed gradually decreases, approaching zero. You never really jump from a high speed to zero instantly—the approach is gradual. Similarly, in mathematics, a limit describes the value a function is approaching as the input gets closer to a particular point, even if the function doesn’t exactly reach that value at the point itself. This Figure 5.2 visualizes how a car approaches a stop sign over time, using the function \(d(t) = 10 e^{-0.3 t}\) to represent the distance to the stop sign.
Explanation of the Graph: \((d(t) = 10 e^{-0.3 t}\)
The graph (Figure 5.2) demonstrates the concept of a mathematical limit, showing how the distance decreases exponentially and approaches zero without ever becoming negative.
- Blue line: shows the car’s distance to the stop sign over time.
- Blue point: represents the car’s position at each moment, moving down along the function.
- Red dashed line: the stop sign, acting as a horizontal asymptote at \(y = 0\).
- Function behavior:
- Exponentially decreasing: the distance drops quickly at first, then gradually approaches zero.
- Always decreasing → the car never moves away from the stop sign.
- Exponentially decreasing: the distance drops quickly at first, then gradually approaches zero.
- Limit: \(\lim_{t \to \infty} d(t) = 0\) → the car approaches the stop sign but never passes it.
5.1.1 Basic Limits
The limit of a function describes the behavior of the function as the input approaches a particular point. Formally, the limit of a function \(f(x)\) as \(x \to a\) is written as:
\[ \lim_{x \to a} f(x) = L \]
This means that as \(x\) gets closer to \(a\), \(f(x)\) gets closer to \(L\).
Example: \[ f(x) = 2x + 3 \] \[ \lim_{x \to 1} f(x) = 2(1) + 3 = 5 \]
Notes: - The limit does not have to equal the function’s value at that point.
- If \(f(a)\) exists, the limit may equal \(f(a)\). If not, the limit can still exist.
5.1.2 One-Sided Limits
Left-hand limit (approaching from the left):
\[ \lim_{x \to a^-} f(x) = L \]
The function approaches \(L\) as \(x\) approaches \(a\) from the left (\(x < a\)).Right-hand limit (approaching from the right):
\[ \lim_{x \to a^+} f(x) = L \]
The function approaches \(L\) as \(x\) approaches \(a\) from the right (\(x > a\)).
Limit exists if:
\[
\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)
\]
Example: \[ f(x) = \begin{cases} x + 2 & x < 1 \\ 3x & x \ge 1 \end{cases} \]
- \(\lim_{x \to 1^-} f(x) = 1 + 2 = 3\)
- \(\lim_{x \to 1^+} f(x) = 3(1) = 3\)
- So, \(\lim_{x \to 1} f(x) = 3\)
5.1.3 Infinite Limits
- Infinite limits occur when the function grows without bound as it approaches a certain point.
\[ \lim_{x \to a} f(x) = \infty \quad \text{or} \quad -\infty \]
Example: \[
f(x) = \frac{1}{(x-2)^2}
\]
\[
\lim_{x \to 2} f(x) = \infty
\]
- This shows the function blows up as \(x\) approaches 2.
5.1.4 Limits at Infinity
- Limits as \(x \to \infty\) or \(x \to -\infty\) describe the end behavior of a function.
Example: \[
f(x) = \frac{3x^2 + 2}{x^2 + 5}
\]
\[
\lim_{x \to \infty} f(x) = 3
\] - Determined by comparing the highest-degree terms in numerator and denominator.