5.2 One-Sided Limits

One-sided limits help us understand how functions behave near tricky points. They're like looking at a function from the left or right side as we get super close to a specific value. This concept is crucial for grasping limits and continuity.

By comparing left-hand and right-hand limits, we can figure out if a function has a regular limit at a point. If both sides match up, we've got a limit. If not, no dice. This idea is super helpful for analyzing piecewise functions and spotting discontinuities.

Left-hand vs Right-hand Limits

Understanding One-sided Limits

• Left-hand and right-hand limits describe the behavior of a function as it approaches a specific point from the left side (values less than the point) or right side (values greater than the point)
• The left-hand limit of a function $f(x)$ as $x$ approaches a point $a$ is denoted as $\lim_{x \to a^-} f(x)$
  • This notation indicates that $x$ approaches $a$ from values less than $a$
• The right-hand limit of a function $f(x)$ as $x$ approaches a point $a$ is denoted as $\lim_{x \to a^+} f(x)$
  • This notation indicates that $x$ approaches $a$ from values greater than $a$
• One-sided limits are useful for determining the behavior of a function near a point of discontinuity or a point where the function is not defined

Relationship between One-sided Limits and Limits

• If the left-hand and right-hand limits of a function at a point are equal, the function is said to have a limit at that point
  • In other words, $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^-} f(x) = L$ and $\lim_{x \to a^+} f(x) = L$
• If either the left-hand or right-hand limit does not exist or if they are not equal, the function does not have a limit at that point
• Example: Consider the function $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$
  • $\lim_{x \to 1^-} f(x) = 2$ and $\lim_{x \to 1^+} f(x) = 2$, so $\lim_{x \to 1} f(x) = 2$

One-sided Limits: Graphical & Numerical

Graphical Method

• To evaluate a one-sided limit graphically, observe the behavior of the function as it approaches the point of interest from the left or right side on the graph
• If the function appears to approach a specific value as $x$ approaches the point from the left (or right), that value is the left-hand (or right-hand) limit
• Example: For the function $f(x) = \begin{cases} x^2, & x < 0 \\ x, & x \geq 0 \end{cases}$, the left-hand limit at $x = 0$ is $0$ and the right-hand limit at $x = 0$ is also $0$

Numerical Method

• To evaluate a one-sided limit numerically, create a table of values for the function as $x$ approaches the point of interest from the left or right side
• Observe the trend in the function values as $x$ gets closer to the point
  • If the function values approach a specific value, that value is the one-sided limit
• When using the numerical method, choose $x$-values that are increasingly close to the point of interest for a more accurate estimate of the one-sided limit
• One-sided limits can be infinite (positive or negative) if the function values tend to positive or negative infinity as $x$ approaches the point from the left or right side
• One-sided limits can be equal to the function value at the point if the function is continuous at that point

Limit Existence: Comparing Sides

Conditions for Limit Existence

• A function has a limit at a point if and only if both the left-hand and right-hand limits exist and are equal
• If the left-hand and right-hand limits are different or if either one-sided limit does not exist, then the function does not have a limit at that point
• To determine the existence of a limit, evaluate both the left-hand and right-hand limits using graphical or numerical methods
  • If the left-hand and right-hand limits are equal and finite, the limit exists and is equal to the common value
  • If the left-hand and right-hand limits are equal and infinite (both positive or both negative), the limit exists and is equal to the corresponding infinity

Examples of Limit Existence and Non-existence

• Example of a function with a limit: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$
  • $\lim_{x \to 1^-} f(x) = 2$ and $\lim_{x \to 1^+} f(x) = 2$, so $\lim_{x \to 1} f(x) = 2$
• Example of a function without a limit: $g(x) = \begin{cases} 1, & x < 0 \\ 0, & x \geq 0 \end{cases}$ at $x = 0$
  • $\lim_{x \to 0^-} g(x) = 1$ and $\lim_{x \to 0^+} g(x) = 0$, so $\lim_{x \to 0} g(x)$ does not exist

Piecewise Functions & One-sided Limits

Analyzing Piecewise-defined Functions

• Piecewise-defined functions are functions that are defined by different expressions or rules for different intervals of the domain
• To analyze the behavior of a piecewise-defined function at a point where the definition changes, evaluate the one-sided limits at that point
• When given a piecewise-defined function, identify the intervals over which each piece of the function is defined and the corresponding expressions or rules
  • Example: $h(x) = \begin{cases} x^2, & x < 1 \\ 2x - 1, & x \geq 1 \end{cases}$
    • For $x < 1$, $h(x) = x^2$
    • For $x \geq 1$, $h(x) = 2x - 1$

Continuity of Piecewise-defined Functions

• To determine the continuity of a piecewise-defined function at a point where the definition changes, compare the left-hand limit, the right-hand limit, and the function value (if defined) at that point
• If the left-hand limit, the right-hand limit, and the function value (if defined) are all equal at a point where the definition changes, the piecewise-defined function is continuous at that point
• Example: For $h(x) = \begin{cases} x^2, & x < 1 \\ 2x - 1, & x \geq 1 \end{cases}$, check continuity at $x = 1$
  • $\lim_{x \to 1^-} h(x) = 1$, $\lim_{x \to 1^+} h(x) = 1$, and $h(1) = 2(1) - 1 = 1$
  • Since the left-hand limit, the right-hand limit, and the function value are all equal at $x = 1$, $h(x)$ is continuous at $x = 1$