Properties of Limits (College Board AP® Calculus AB): Revision Note

Properties of limits

What properties of limits do I need to know?

• There are a number of limit properties (also known as limit theorems) that you need to know and be able to use

  • They let you work out the limits of complicated functions algebraically by combining the limits of simpler functions

• The limit of a constant function: If $k$ is a constant then

  • $\underset{x\to a}{\lim }k=k$

• The limit of a multiple of a function: If$k$ is a constant and $\underset{x\to a}{\lim }f\left(x\right)=L$, then

  • $\underset{x\to a}{\lim }\left(kf\left(x\right)\right)=kL$

• The limit of a sum or difference of functions: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$, then

  • $\underset{x\to a}{\lim }\left(f\left(x\right)\pm g\left(x\right)\right)=L\pm M$

• The limit of a product of functions: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$, then

  • $\underset{x\to a}{\lim }\left(f\left(x\right)·g\left(x\right)\right)=L·M$

• The limit of a quotient of functions: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$ with $M\ne 0$, then

  • $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{L}{M}$

• The limit of the power of a function: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $n$ is a real number, then

  • $\underset{x\to a}{\lim }{\left[f\left(x\right)\right]}^n=L^n$

• The limit of a composite function: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and if the function $g$ is continuous at $x=L$, then

  • $\underset{x\to a}{\lim }g\left(f\left(x\right)\right)=g\left(L\right)$

• Note that statements like $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$ assume that those limits exist, and that $L$ and $M$ are real numbers

How do the properties of limits work with infinite limits?

• There are several properties of limits involving infinite limits

  • They help determine whether a function increases without bound (i.e. tends to $\infty$) or decreases without bound (i.e. tends to $-\infty$) at a particular point

• Limit of $\frac{\mathbf{1}}{{\mathit{x}}^{\mathit{n}}}$ at zero: If $n$ is a positive integer, then

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

  • $\underset{x\to 0^{-}}{\lim }\frac{1}{x^n}=\{\begin{array}{l}\infty \mathrm{if}n\mathrm{is}\mathrm{even}\\ -\infty \mathrm{if}n\mathrm{is}\mathrm{odd}\end{array}$

• Infinite limits of quotients: If $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=0$, then

  • if $L>0$, $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\{\begin{array}{l}\infty ,\mathrm{if}g\left(x\right)>0\mathrm{as}x\mathrm{approaches}a\\ -\infty ,\mathrm{if}g\left(x\right)<0\mathrm{as}x\mathrm{approaches}a\end{array}$

  • if $L<0$, $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\{\begin{array}{l}-\infty ,\mathrm{if}g\left(x\right)>0\mathrm{as}x\mathrm{approaches}a\\ \infty ,\mathrm{if}g\left(x\right)<0\mathrm{as}x\mathrm{approaches}a\end{array}$

  • In both these cases, the limits from the left (as $x\to a^{-}$) and right (as $x\to a^{+}$) may be different

    • Check the behavior of $g\left(x\right)$ to determine the correct limit