Introduction to Limits (DP IB Analysis & Approaches (AA)): Revision Note

Limits

What are limits in mathematics?

• When we consider a limit in mathematics we look at the tendency of a mathematical process as it approaches, but never quite reaches, an ‘end point’ of some sort

• We use a special limit notation to indicate this

  • For example  denotes ‘the limit of the function f(x) as x goes to (or approaches) 3’

    • I.e., what value (if any) f(x) gets closer and closer to as x takes on values closer and closer to 3

    • We are not concerned here with what value (if any) f(x) takes when x is equal to 3 – only with the behaviour of f(x) as x gets close to 3

• The sum of an infinite geometric sequence is a type of limit

  • When you calculate for an infinite geometric sequence, what you are actually finding is 

    • I.e., what value (if any) the sum of the first n terms of the sequence gets closer and closer to as the number of terms (n) goes to infinity

    • The sum never actually reaches , but as more and more terms are included in the sum it gets closer and closer to that value

• In this section of the IB course we will be considering the limits of functions

  • This may include finding the limit at a point where the function is undefined

    • For example,  is undefined when x = 0, but we might want to know how the function behaves as x gets closer and closer to zero

  • Or it may include finding the limit of a function f(x) as x gets infinitely big in the positive or negative direction

    • For this type of limit we write  or  (the first one can also be written as  to distinguish it from the second one)

  • These sorts of limits are often used to find the asymptotes of the graph of a function

How do I find a simple limit?

• STEP 1: To find  begin by substituting a into the function f(x)

  • If f(a) exists with a well-defined value, then that is also the value of the limit

  • For example, for  we may find the limit as x approaches 3 like this:

    
    
    

    • In this case,  is simply equal to f(3)

• STEP 2: If f(a) does not exist, it may be possible to use algebra to simplify f(x) so that substituting a into the simplified function gives a well-defined value

  • In that case, the well-defined value at x = a of the simplified version of the function is also the value of the limit of the function as x goes to a

  • For example,  is not defined at x = 0, but we may use algebra to find the limit as x approaches zero:
    
    
    

    • Note that  and  are not the same function!

    • They are equal for all values of x except zero

    • But for x = 0, g(0) = 0 while f(0) is undefined

    • However f(x) gets closer and closer to zero as x gets closer and closer to zero

• If neither of these steps gives a well-defined value for the limit you may need to consider more advanced techniques to evaluate the limit

  • For example l’Hôpital’s Rule or using Maclaurin series

How do I find a limit to infinity?

• As x goes to  or , a typical function f(x) may converge to a well-defined value, or it may diverge to or 

  • Other behaviours are possible – for example  is simply undefined, because sin x continues to oscillate between 1 and -1 as x gets larger and larger

• There are two key results to be used here:

  •   converges to 0 for all n >0 and all 

  •  diverges to  for all n > 0

    •  for n > 0 will need to be considered on a case-by-case basis, because of the differing behaviour of xⁿ for different values of n when x is negative

• STEP 1: If necessary, use algebra to rearrange the function into a form where one or the other of the key results above may be applied

• STEP 2: Use the key results above to evaluate your limit

• For example:

• Or:
  
  
  

  • I.e., the limit diverges to (because  it gets bigger and bigger without limit as x gets bigger and bigger)

• Remember that neither  nor  has a well-defined value!

  • If you attempt to evaluate a limit and get one of these two forms, you will need to try another strategy

  • This may just mean different or additional algebraic rearrangement

  • But it may also mean that you need to consider using l’Hôpital’s Rule or Maclaurin series to evaluate the limit

• It is also worth remembering that if , then for any non-zero 

  • This can be useful for example when evaluating the limits of functions containing exponentials

    •  for any p > 0, so we immediately have  for p > 0

    • See the worked example below for a more involved version of this

Do limits ever have ‘directions’?

• Yes they do!

• The notation means ‘the limit of f(x) as x approaches a from above’

  • I.e., this is the limit as x comes ‘down’ towards a, only considering the function’s behaviour for values of x that are greater than a

• The notation means ‘the limit of f(x) as x approaches a from below’

  • I.e., this is the limit as x comes ‘up’ towards a, only considering the function’s behaviour for values of x that are less than a

• One place these sorts of limits appear is for functions defined piecewise

  • In this case the limits ‘from above’ and ‘from below’ may well be different for values of x at which the different ‘pieces’ of the function are joined

• But also be aware of a situation like the following, where the limits from above and below may also be different:

  •  (because  for x > 0, with  becoming bigger and bigger in the positive direction as x gets closer and closer to zero ‘from above’)

  •  (because  for x < 0, with  becoming bigger and bigger in the negative direction as x gets closer and closer to zero ‘from below’)

  • The graph of  shows this limiting behaviour as x approaches zero from the two different directions