Mathematics 265 Introduction to Calculus I

Study Guide :: Unit 3

Limits

Evaluation of the limit ${\lim\limits_{x \to \infty }} f(x)$

Is $f$ a polynomial function?		$\overset{YES}{⟶}$		The limit is $\infty$ if $f (x) > 0$ for $x \to \pm \infty$ ; it is $- \infty$ if $f (x) < 0$ for $x \to \pm \infty .$
$↓ NO$
Is $f$ a rational function?		$\overset{YES}{⟶}$		Apply Theorem 3.29.
$↓ NO$
Is $f$ the quotient of radicals of polynomial functions?		$\overset{YES}{⟶}$		Simplify and apply Proposition 3.16 and Theorem 3.23.
$↓ NO$
Is $f (x) = h (g (x))$ for $x$ near $a$ , $\lim_{x \to \infty} g (x) = b$ and $\lim_{x \to b} h (x) = \pm \infty ?$		$\overset{YES}{⟶}$		See Theorem 3.20.
$↓ NO$
Are there functions $g$ and $h$ such that $g (x) \leq f (x) \leq h (x)$ for $x \to \infty$ and $\lim_{x \to \infty} h (x) = \lim_{x \to \infty} g (x) ?$		$\overset{YES}{⟶}$		Apply the Squeeze Theorem.
$↓ NO$
Other techniques may be required; for example, L’Hospital’s Rule or power series, which are studied in more advanced courses.

The same flow chart can be used for the limits $\lim_{x \to + \infty} f (x)$ and $\lim_{x \to - \infty} f (x) .$

For help with the concept of limits, view the PowerPoint tutorials below. To access a tutorial:

Click on the file to open it.
Extract the files and save them to your computer’s hard drive.
Click on the file folder to open it.
Click on the PowerPoint file to view and listen.

Note: If you don’t have PowerPoint installed on your computer, you can download the PowerPoint Viewer from here: https://support.microsoft.com/en-us/office/view-a-presentation-without-powerpoint-2f1077ab-9a4e-41ba-9f75-d55bd9b231a6?ui=en-us&rs=en-us&ad=us.

Tutorial 5: Limits

Tutorial 6: Limits

Tutorial 7: Limits