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

Limits

What are limits in mathematics?

• A limit in mathematics involves 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 $\underset{x\to 3}{\lim }f\left(x\right)$ denotes ‘the limit of the function f(x) as x goes to (or approaches) 3’

    • I.e., what value (if any) does f(x) get 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 $S_{\infty }$ for an infinite geometric sequence, you are actually finding $\underset{n\to \infty }{\lim }{S}_n$

    • 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 $S_{\infty }$

      • but as more and more terms are included in the sum it gets closer and closer to that value

What can I be asked to do with limits?

• In the IB course you will normally be considering the limits of functions

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

    • For example, $f\left(x\right)=\frac{\sin x}{x}$ is undefined when x = 0

      • but you 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 $\underset{x\to \infty }{\lim }f\left(x\right)$ or $\underset{x\to -\infty }{\lim }f\left(x\right)$ 

      • (The first one can also be written as $\underset{x\to +\infty }{\lim }f\left(x\right)$ 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 $\underset{x\to a}{\lim }f\left(x\right)$ 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

    • E.g., for $f\left(x\right)=\frac{x-1}{x}$

      • $\underset{x\to 3}{\lim }f\left(x\right)=\underset{x\to 3}{\lim }\frac{x-1}{x}=\frac{3-1}{3}=\frac{2}{3}$

    • In this case, $\underset{x\to 3}{\lim }f\left(x\right)$ is simply equal to f(3)

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

  • In that case, the value of the simplified version of the function is also the value of the limit of the function as x goes to a

    • E.g. $f\left(x\right)=\frac{x^2}{x}$ is not defined at x = 0, but you may use algebra to find the limit as x approaches zero

      • $\underset{x\to 0}{\lim }f\left(x\right)=\underset{x\to 0}{\lim }\frac{x^2}{x}=\underset{x\to 0}{\lim }\frac{x}{1} \left(\mathrm{cancelling}\mathrm{ }\mathrm{the}\mathrm{ }{x}^{\mathrm{\text{'}}}\mathrm{s}\right)=\frac{0}{1}=0$

    • Note that $f\left(x\right)=\frac{x^2}{x}$ and $g\left(x\right)=x$ 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 $+\infty$ or $-\infty$, a function f(x) may converge to a well-defined value, or it may diverge to $+\infty$ or $-\infty$

  • Other behaviours are possible

    • For example $\underset{x\to \mathrm{\infty }}{\lim }\sin x$ 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:

  • $\underset{x\to \pm \mathrm{\infty }}{\lim }\frac{k}{x^n}$  converges to 0 for all n >0 and all $k\in \mathrm{ℝ}$

  • $\underset{x\to +\mathrm{\infty }}{\lim }{x}^n$ diverges to $+\infty$ for all n > 0

    • $\underset{x\to -\mathrm{\infty }}{\lim }{x}^n$ for n > 0 will need to be considered on a case-by-case basis

      • xⁿ behaves differently 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

  • $\underset{x\to \infty }{\lim }\frac{3x^2-2x+1}{4x^2-x+2}=\underset{x\to \infty }{\lim }\frac{3-\frac{2}{x}+\frac{1}{x^2}}{4-\frac{1}{x}+\frac{2}{x^2}}=\frac{3-0+0}{4-0+0}=\frac{3}{4}$

• Or

  • $\underset{x\to +\infty }{\lim }\frac{x^2+5x-2}{32x+3}=\underset{x\to +\infty }{\lim }\frac{x+5-\frac{2}{x}}{32+\frac{3}{x}}=\frac{\left(+\mathrm{\infty }\right)+5-0}{32+0}=+\mathrm{\infty }$
    

    • I.e., the limit diverges to $+\infty$ (because $\frac{x^2+5x-2}{32x+3}$ gets bigger and bigger without limit as x gets bigger and bigger)

• Another useful result is that if $\underset{x\to \infty }{\lim }f\left(x\right)=\mathrm{\infty }$, then $\underset{x\to \infty }{\lim }\frac{k}{f\left(x\right)}=0$ for any $k\in \mathrm{ℝ}$

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

    • E.g. $\underset{x\to \infty }{\lim }{\mathrm{e}}^{px}=\mathrm{\infty }$ for any p > 0,

      • So you immediately know $\underset{x\to \infty }{\lim }{\mathrm{e}}^{-px}=\underset{x\to \infty }{\lim }\frac{1}{{\mathrm{e}}^{px}}=0$ 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 $\underset{x\to a^{+}}{\lim }f\left(x\right)$ means ‘the limit of f(x) as x approaches a from above’

  • I.e., this is the limit as x comes ‘down’ towards a

    • It only considers the function’s behaviour for values of x that are greater than a

• The notation $\underset{x\to a^{-}}{\lim }f\left(x\right)$ means ‘the limit of f(x) as x approaches a from below’

  • I.e., this is the limit as x comes ‘up’ towards a

    • It only considers 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

  • Limits ‘from above’ and ‘from below’ may well be different at values of x where the different ‘pieces’ of the function are joined

• But also be aware of a situation like the following:

  • $\underset{x\to 0^{+}}{\lim }\frac{1}{x}=+\mathrm{\infty }$ 

    • Because $\frac{1}{x}>0$ for x > 0, with $\frac{1}{x}$ becoming bigger and bigger in the positive direction as x gets closer and closer to zero ‘from above’

  • $\underset{x\to 0^{-}}{\lim }\frac{1}{x}=-\mathrm{\infty }$

    • Because $\frac{1}{x}<0$ for x < 0, with $\frac{1}{x}$ becoming bigger and bigger in the negative direction as x gets closer and closer to zero ‘from below’

  • The graph of $y=\frac{1}{x}$ shows this limiting behaviour as x approaches zero from the two different directions