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 .
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 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 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 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 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 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