Limits

Finishing This Unit

Review the objectives of this unit and make sure you are able to meet all of them.
If there is a concept, definition, example or exercise that is not yet clear to you, go back and reread it, then contact your tutor for help.
Diagrams summarizing strategies for evaluating limits are presented below. Note that these strategies are recommendations—with more practice, you may develop your own strategies. Until then, you may wish to detach these diagrams and pin them above your desk for easy reference.
Do the exercises in “Learning from Mistakes” section for this unit.
You may want to do Exercises 1-16, 18, 23 parts (a) and (b) and 24-26 from the “Review” (pages 98-99 of the textbook).

Is $f$ continuous at $a ?$	$\overset{YES}{⟶}$	By Theorem 3.14, $\lim_{x \to a} f (x) = f (a) .$
$↓ NO$
Can $f (x)$ be simplified so that $f (x) = g (x)$ for $x$ near $a$ , and $\lim_{x \to a} g (x) = M$	$\overset{YES}{⟶}$	By Proposition 3.16, $\lim_{x \to a} f (x) = \lim_{x \to a} g (x) = M .$
$↓ NO$
Is $f$ equal to, or can it be simplified to a sum, product or quotient of functions each of which has finite limits at $a ?$	$\overset{YES}{⟶}$	Theorem 3.23 may apply. If it is a quotient, check that the denominator limit is nonzero.
$↓ NO$
Is $f (x) = h (g (x))$ for $x$ near $a$ , $\lim_{x \to a} g (x) = b$ , and either $\lim_{x \to b} h (x) = \pm \infty$ or $\lim_{x \to b} h (x) = L ?$	$\overset{YES}{⟶}$	See Theorems 3.20 and 3.24, and Corollaries 3.25 and 3.26.
$↓ NO$
Is $f (x) = h (x) g (x)$ with $\lim_{x \to a} g (x) = L$ ( $L \neq 0$ ) and $\lim_{x \to a} h (x) = \pm \infty ?$	$\overset{YES}{⟶}$	See Theorem 3.18.
$↓ NO$
Are there functions $g$ and $h$ such that $g (x) \leq f (x) \leq h (x)$ near $a$ and $\lim_{x \to a} h (x) = \lim_{x \to a} g (x) ?$	$\overset{YES}{⟶}$	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 left and right limits: $\lim_{x \to a^{+}} f (x)$ and $\lim_{x \to a^{-}} f (x) .$