Determining Limits

There are two conditions to determining limits by inspection

1. x goes to positive or negative infinity

2. limit involves a polynomial divided by a polynomial

Example:

lim (x³ - 4)/(x² +x+1)

This problem on two conditions:

1. polynomial over polynomial

2. x approaches infinity

If the highest power of x is greater in numerator so the limit is positive or negative infinity

Example:

1. lim (x³ - 4)/(x² +x+1)

- Highest power of x in numerator is 3

- Highest power of x in denominator is 2

Since all the number are positive and x going to positive infinity so value the limit is infinity.

lim (x³ - 4)/(x² +x+1) = infinity

If you can’t tell if the answer is positive or negative infinity:

Ø You can substitute a large number for x

Ø See if you end up with a positive or negative number

Ø Whatever sign you get is the sign of infinity for the limit

2. lim (x³ - 4)/(x⁴ +3x+5)

- Highest power of x in numerator is 3

- Highest power of x in denominator is 4

lim (x³ - 4)/(x⁴ +3x+5)

= ((x³-4)/x⁴)/(x⁴+3x+5)/x⁴)

= 0