Unit 1 Summary (College Board AP® Calculus BC): Revision Note

Limits & continuity summary

Key facts & definitions

• The limit of a function $f$ at the value $x=c$ is $\underset{x\to c}{\lim }f\left(x\right)=L$ if

  • if $\underset{x\to c^{-}}{\lim }f\left(x\right)$ and $\underset{x\to c^{+}}{\lim }f\left(x\right)$ exist

  • $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)=L$

• A function $f$ is continuous at the value $x=c$ if

  • $f\left(c\right)$ and $\underset{x\to c}{\lim }f\left(x\right)$ exist

  • and $\underset{x\to c}{\lim }f\left(x\right)=f\left(c\right)$

• A function $f$ has a removable discontinuity at the value $x=c$ if

  • $f\left(c\right)$ and $\underset{x\to c}{\lim }f\left(x\right)$ exist

  • and $\underset{x\to c}{\lim }f\left(x\right)\ne f\left(c\right)$

• $y=c$ is a horizontal asymptote for the graph $y=f\left(x\right)$ if $\underset{x\to \infty }{\lim }f\left(x\right)=c$ or $\underset{x\to -\infty }{\lim }f\left(x\right)=c$

• $x=c$ is a vertical asymptote for the graph $y=f\left(x\right)$ if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\pm \infty$ or $\underset{x\to c^{+}}{\lim }f\left(x\right)=\pm \infty$

Key theorems

• Squeeze theorem

  • Let $f$, $g$ and $h$ be functions defined on an open interval including $a$ such that

    • $g\left(x\right)\le f\left(x\right)\le h\left(x\right)$ for all $x$ in the interval (except possibly $a$), and

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

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

• Intermediate value theorem (IVT)

  • If $f$ is a continuous function on the closed interval $\left[a,b\right]$

  • and if $d$ is a value within the closed interval created by $f\left(a\right)$ and $f\left(b\right)$

  • then there is at least one number $c$ between $a$ and $b$ such that $f\left(c\right)=d$

Key formulas

• 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

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