1.5 Determining Limits Using Algebraic Properties of Limits

Learning objectives

—LIM-1.D—Determine the limits of functions using limit theorems• LIM-1.D Determine the limits of functions using limit theorems.• LIM-1.D.1 One-sided limits can be determined analytically or graphically.• LIM-1.D.2 Limits of sums, differences, products, quotients, and composite functions can be found using limit theorems.• Enduring understanding LIM-1: Reasoning with definitions, theorems, and properties can be used to justify claims about limits.

Limit theorems let you find the limit of a compound expression from the limits of its parts. First identify the outer operation, then verify that the required component limits and any extra conditions exist.

\text{If }\lim_{x\to c}f(x)=A\text{ and }\lim_{x\to c}g(x)=B,\text{ then:}

Expression	Limit	Required condition
$f(x)\pm g(x)$	$A\pm B$	Both component limits exist
$f(x)g(x)$	$AB$	Both component limits exist
$\dfrac{f(x)}{g(x)}$	$\dfrac{A}{B}$	$B\ne0$
$h(g(x))$	$h(B)$	$h$ is continuous at $B$

For $\displaystyle\lim_{x\to2}\frac{(3x-1)(x+4)}{x+1}$ , the component limits are $5$ , $6$ , and $3$ . Because the denominator limit is $3\ne0$ , the product and quotient theorems apply: $\displaystyle\frac{(5)(6)}{3}=10$ .

The same laws apply to $x\to c^-$ or $x\to c^+$ when every component limit is taken from that same side. A two-sided result is justified only when the final left-hand and right-hand limits agree.

Do not use the quotient theorem when the denominator limit is $0$ , and do not assume that combining expressions repairs a missing component limit. In either case, a different analysis is needed before a conclusion can be made.

Next Topic

—1.6 Determining Limits Using Algebraic ManipulationRead