Selecting Procedures for Determining Limits (College Board AP® Calculus AB)

Study Guide

Written by: Roger B

Reviewed by: Dan Finlay

Selecting procedures for determining limits

How do I choose the correct procedure for evaluating a limit?

You should be familiar with all the different methods for estimating or evaluating limits
- This way you can choose the most appropriate method to use for an exam question
You can estimate the value of a limit using
- A table of values for a function
- The graph of a function
  - The graph may be given in the question, or produced on your graphing calculator
- See the 'Evaluating Limits Numerically & Graphically' study guide
- An estimate is not enough to absolutely determine the value of a limit
  - But it can be used to check limits that you determine analytically
You can evaluate a limit analytically using
- Substitution
  - For a function continuous on an interval including the point in question
  - For limits at the 'joins' of piecewise functions
- Simplifying algebraically
  - For quotient functions where the numerator and/or denominator can be factorised and common factors canceled
- Multiplying by a conjugate
  - For quotient functions involving surds, e.g. $\begin{array}{rcl} \frac{\sqrt{x + 5} - \sqrt{5}}{x} \end{array}$
- Multiplying by a reciprocal
  - For quotient functions involving polynomials, e.g. $\begin{array}{rcl} \frac{7 x^{2} - 4 x + 13}{2 x^{2} + 3 x - 5} \end{array}$
- See the 'Evaluating Limits Analytically' study guide
You can also evaluate a limit using
- The squeeze theorem
  - For a function bounded above and below by two other functions whose limits are equal at a point
- Trigonometric limit theorems
  - Using the results $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{\cos x - 1}{x} = 0$ along with algebraic manipulation and properties of limits
- See the 'Squeeze Theorem & Trigonometric Limits' study guide

Examiner Tips and Tricks

Make sure you are familiar with the properties of limits

These let you evaluate more complicated limits in terms of more basic limits
See the 'Properties of Limits' study guide

Are there any hidden tricks for evaluating limits?

Remember that certain processes in calculus are defined using limits
- If you recognise a limit as matching one of these definitions, you may be able to use the process to find the value of the limit
The derivative of a function $f$ is defined as $f^{'} (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$ (see the 'Derivatives & Tangents' study guide)
- Consider $\lim_{t \to 0} \frac{\sin (\frac{π}{6} + t) - \frac{1}{2}}{t}$
  - $\sin (\frac{π}{6}) = \frac{1}{2}$ so this is equivalent to $\lim_{t \to 0} \frac{\sin (\frac{π}{6} + t) - \sin (\frac{π}{6})}{t}$
  - That's the definition of the derivative of $\sin x$ at $x = \frac{π}{6}$ (just using $t$ instead of $h$ )
  - ${\frac{d}{d x} (\sin x)}_{x = \frac{π}{6}} = {\cos x}_{x = \frac{π}{6}} = \cos (\frac{π}{6}) = \frac{\sqrt{3}}{2}$
  - So $\lim_{t \to 0} \frac{\sin (\frac{π}{6} + t) - \frac{1}{2}}{t} = \frac{\sqrt{3}}{2}$
The definite integral of a function $f$ is defined as the limit of Riemann sums $\int_{a}^{b} f (x) d x = \lim_{\max Δ x_{i} \to 0} \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x_{i}$ (see the 'Properties of Definite Integrals' study guide)
- Consider $\lim_{\max Δ x_{i} \to 0} \sum_{i = 1}^{n} \cos (x_{i}^{*}) Δ x_{i}$ over the interval between $x = 0$ and $x = \frac{π}{2}$
  - That is just the definition of $\int_{0}^{\frac{π}{2}} \cos x d x$
  - And $\int_{0}^{\frac{π}{2}} \cos x d x = 1$
  - So over the interval between $x = 0$ and $x = \frac{π}{2}$ , $\lim_{\max Δ x_{i} \to 0} \sum_{i = 1}^{n} \cos (x_{i}^{*}) Δ x_{i} = 1$

Last updated: 1 August 2024

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