A limit is the value that a mathematical function approaches to as its input approaches some value. In other words, it is the value that a function converges to, but does not necessarily equate to. Limits are used to define derivates (using First Principles) and continuity, as well as to find the long-term behaviour of a function
Definition & Properties
Let , where , the domain, is a subset of all real numbers (i.e. a real-valued function). Then:
- As x approaches a, f(x) approaches (or converges) to L
Well, that’s a simplified definition. A more formal (and mathematically correct definition) can be found here: Formal Definition of a Limit
Limits must be:
- Unique: There cannot be two limits for a given function at the same point. This leads into the > Theorem - Limits from different Sides
- Real: Limits must belong to the real set,
- Finite: While x can approach infinity, the limit of must be finite i.e. limits can never be infinity. This property makes solving limits tricky, which is why various methods to evaluate limits exist
Theorem - Limits from different Sides
Since limits must be unique, piecewise functions cannot have a defined limit if, as the input approaches a value, the function approaches multiple values, even if they aren’t defined at that input value. I.e. cannot equal both and (or however many possible values)
Different Limits?
Take the following graph of the function:
What is ? Is it 1 or -1?
As you can see, limits can have different values if the function is approached from the right or left side. Hence, to solve this problem a simple theorem exists:
In plain words, the limit exists at a point a if, and only if, the right hand and left hand side limits both exist and are equal. In the example above, because (limit from the right side) and (limit from the left side), the limits from different sides are unequal. Hence, the limit does not exist at .
Limit Laws
Limit Laws
Just as most operators, like the integral and have their own respective laws as to what addition, multiplication, etc. mean for them, limits also have certain limit laws. These help decompose a complex limit into a simpler version.
Laws
Addition does not affect limits, we can ‘pull out’ the plus sign from the limit. E.g.
Constant factors multiplication (and division) do not affect limits. E.g.
Link to source
Standard Limits
Standard Limits
The following limits do need any further proof to be used. Make sure to say that they are standard limits in working out.
Link to source
Solving Limits As X Approaches Infinity
There are multiple methods for solving limits as x approaches infinity (since limits cannot equal infinity) that have unique conditions.
Function Form | Technique | Notes |
---|---|---|
where common terms exist | Divide numerator and denominator by the [[Complexity Analysis#asymptotic-notation | fastest growing term]]. |
Multiply and divide by conjugate | If the function is expressed as then it’s conjugate is | |
or | [[L’Hôpital’s Rule | L’Hôpital’s Rule]] |
None of the previous methods work | [[Sandwich Theorem | Sandwich Theorem]] |