99. Limits
99.0.1. Properties of Limits
99.0.1.1. Continuous
99.0.1.1.1. Addition, Subtraction, Multiplication, Division
99.0.1.1.2. Constant
99.0.1.2. Non-continuous
Even though the limit for either function may not exist, their can exist as long as
Example
Problem 1: Limit Exists
- Left-hand limit ()
Adding these
- Right-hand limit ()
Adding these
- Since both the left-hand and right-hand limits agree
Problem 2: Limit Does Not Exist
- Left-hand limit ()
Adding these
- Right-hand limit ()
Adding these
- Since both the left-hand and right-hand limits do not agree, the limit does not exist
99.0.1.3. Composite Functions
For this to hold true, two important conditions must be satisfied:
-
Inner limit exists: The limit must exist and equal some value . That is, as gets arbitrarily close to , approaches a well-defined number
-
Continuity of the outer function: The function must be continuous at the point . Continuity ensures that behaves predictably near , without any jumps, gaps, or undefined points
Example
Problem 1: Inner Limit & Continuity Exist
- Inner limit
Observing , as , . The inner limit exists
- Continuity of at
Observing , . Since is continuous at , the composite limit holds
Problem 2: Inner Limit Does Not Exist
- Inner limit
Observing , as , . The inner limit does not exist
Problem 3: Continuity Does Not Exist
- Inner limit
Observing , as , . The inner limit exists
- Continuity of at
Observing , is not continuous. Since is not continuous at , the composite limit does not hold
99.0.1.4. Limits by Direct Substitution
Example
Limit exists
Limit does not exist (Undefined)
99.0.1.4.1. Limits of Piecewise Functions
99.0.1.4.2. Absolute Value
99.0.1.5. Limits by Factoring
99.0.1.6. Limits by Rationalizing
99.0.1.7. Continuity & Differentiability at a Point
Example
Piecewise function:
- Check for Continuity
- Value of
- Left-Hand Limit (LHL)
- Right-Hand Limit (RHL)
Since , is continuous at
- Check for Differentiability
- Left-Hand Derivative (LHD)
- Right-Hand Derivative
Since the left-hand and right-hand derivatives are equal, is differentiable at
Conclusion: is both continuous & differentiable at