# Calculating limits using algebra

Calculating Limits Using Algebra
• Use of Direct Substitution
• $\lim_{\ x\to c}{F(x)}=F(c)$
This idea states that this function is continuous at whatever value c may be

Some Basic Limits
- Let b and c be real numbers and let n be a positive integer.

• $\lim_{\ x\to c}{b}=b$
• $\lim_{\ x\to c}{x}=c$
• $\lim_{\ x\to c}{x^n}=c^n$

Examples:
$\lim_{\ x\to2}{3}=3$

$\lim_{\ x\to 6}{x}=6$

$\lim_{\ x\to5}{x^2}=5^2=25$

Properties of Limits
-Let b and c be real numbers, let n be a positive integer. and let f and g be functions with the following limits

• $\lim_{\ x\to c}{f(x)}=L$

• $\lim_{\ x\to c}{g(x)}=K$

1.Scalar multiple:
$\lim_{\ x\to c}{bf(x)}=bL$
2. Sum or difference:

$\lim_{\ x\to c}{f(x)\pm g(x)}=L\pm K$

3. Product

$\lim_{\ x\to c}{f(x)g(x)}=LK$

4. Quotient

$\lim_{\ x\to c}\frac{f(x)}{g(x)}=\frac{L}{K}$
K cannot equal 0

5. Power

$\lim_{\ x\to c}{f(x)^n}=L^n$

Example of a Limit of a Polynomial
$\lim_{\ x\to3}{5x^3+5}$

The Limit of a Rational Function
$\lim_{\ x\to -1}\frac{x^2+x+2}{(x+1)}$

Limits of Trigonometric Functions
1. $\lim_{}{sin(x)}=sin(c)$
2. $\lim_{}{cos(x)}=cos(c)$
3. $\lim_{}{tan(x)}=tan(c)$
4. $\lim_{}{cot(x)}=cot(c)$
5. $\lim_{}{sec(x)}=sec(c)$
6. $\lim_{}{csc(x)}=csc(c)$

Functions That Agree at All But One Point

$\lim_{\ x\to 1}\frac{x^3-1}{(x-1)}$

Notice that if you apply the limit as is the denominator will equal 0 which is not allowed. You must factor out the numerator to solve this limit
$f(x)=\frac{(x-1)(x^2+x+1)}{(x-1)}$

Cancel out the (x-1) from the top and bottom by division and all you are left with is
$\lim_{\ x\to 1}{x^2+x+1}$

Source--- Calculus with Analytic Geometry Eighth Edition. Houghton Mifflin Company, Boston/New York
example calculating limits using algebra