Selecting Procedures for Determining Limits (College Board AP® Calculus BC): Study Guide

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. $\begin{array}{rcl}& & \frac{7x^2-4x+13}{2x^2+3x-5}\end{array}$

  • 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 $\underset{x\to 0}{\lim }\frac{\sin x}{x}=1$ and $\underset{x\to 0}{\lim }\frac{\cos x-1}{x}=0$ along with algebraic manipulation and properties of limits

  • See the 'Squeeze Theorem & Trigonometric 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 $f^{\text{'}}\left(x\right)=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$(see the 'Derivatives & Tangents' study guide)

  • Consider $\underset{t\to 0}{\lim }\frac{\sin \left({\displaystyle \frac{\pi }{6}}+t\right)-{\displaystyle \frac{1}{2}}}{t}$

    • $\sin \left(\frac{\pi }{6}\right)=\frac{1}{2}$ so this is equivalent to $\underset{t\to 0}{\lim }\frac{\sin \left({\displaystyle \frac{\pi }{6}}+t\right)-\sin \left({\displaystyle \frac{\pi }{6}}\right)}{t}$

    • That's the definition of the derivative of $\sin x$ at $x=\frac{\pi }{6}$ (just using $t$ instead of $h$)

    • ${\overline{)\frac{d}{dx}\left(\sin x\right)}}_{x=\frac{\pi }{6}}={\overline{)\cos x}}_{x=\frac{\pi }{6}}=\cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}$

    • So $\underset{t\to 0}{\lim }\frac{\sin \left({\displaystyle \frac{\pi }{6}}+t\right)-{\displaystyle \frac{1}{2}}}{t}=\frac{\sqrt{3}}{2}$

• The definite integral of a function $f$ is defined as the limit of Riemann sums ${\int }_a^{b}f\left(x\right)dx=\underset{\max \mathrm{\Delta }{x}_i\to 0}{\lim }\sum _{i=1}^{n}f\left(x_i^{\ast }\right)\mathrm{\Delta }{x}_{\mathit{i}}$ (see the 'Properties of Definite Integrals' study guide)

  • Consider $\underset{\max \mathrm{\Delta }{x}_i\to 0}{\lim }\sum _{i=1}^n\cos \left(x_i^{\ast }\right)\mathrm{\Delta }{x}_i$ over the interval between $x=0$ and $x=\frac{\pi }{2}$

    • That is just the definition of ${\int }_0^{\frac{\pi }{2}}\cos xdx$

    • And ${\int }_0^{\frac{\pi }{2}}\cos xdx=1$

    • So over the interval between $x=0$ and $x=\frac{\pi }{2}$, $\underset{\max \mathrm{\Delta }{x}_i\to 0}{\lim }\sum _{i=1}^n\cos \mathit{\left(}{\mathit{x}}_{\mathit{i}}^{\mathit{\ast }}\mathit{\right)}\mathrm{\Delta }{x}_i=1$