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 limx→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 limx→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, limx→af(x)=L if and only if limx→a−f(x)=L and limx→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)=x−1x2−1 at x=1
- limx→1−f(x)=2 and limx→1+f(x)=2, so limx→1f(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)={x2,x,x<0x≥0, 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)=x−1x2−1 at x=1
- limx→1−f(x)=2 and limx→1+f(x)=2, so limx→1f(x)=2
- Example of a function without a limit: g(x)={1,0,x<0x≥0 at x=0
- limx→0−g(x)=1 and limx→0+g(x)=0, so limx→0g(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)={x2,2x−1,x<1x≥1
- For x<1, h(x)=x2
- For x≥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)={x2,2x−1,x<1x≥1, check continuity at x=1
- limx→1−h(x)=1, limx→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