Describing+asymptotic+behavior+in+terms+of+limits+involving+infinity

Since limits are the range values a function approaches as a domain value becomes close to a range value from both the left and the right, it makes sense that this range value can also be infinity.

Only one limit approaching infinity or negative infinity, from the left or right or both, must occur for there to be a vertical asymptote. To find out wheter or not the vertical asymptote is approaching infinity or negative infinity, simply make a table of values from the right and from the left becoming very close to the domain value of which the limit is being found. If those values are massive then the limit from that direction is to infinity. If the values are massivly negative then the limit is to negative infinity. If the limit from the left eauals the limit from the right, then the limit when approaching the input falue is infinity or negative infinity.

For a horizontal asymptote, the range value approaches a single value as the domain approaches either infinity or negative infinity. To test for a horizontal asymptote, plug in a massive domain value or massivly negative domain value followed by an even more massive version of both. If the numbers are close then the range approaches that value as the range approaches infinity.

End behavior also involves infinite values. If the leading coefficant is positive, then the function approaches infinity as the domain approaches infinity. If the leading coefficant is negative, then the range approaches negative infinity as the domain approaches infinity. Conversely, if the index is even, the the range value will approach the same sign infinity it did when the domain appoached infinity as the range will when the domain approaches negative infinity.

Example 1: What are the limits as x approaches infinity and negative infinity for the function f(x)=4x^4+3x+7

Answer: As x approaches infinity f(x) approaches intinity. As x approaches negative infinity f(x) approaches infinity.

Example 2: What is the horizontal asymptote?

Answer: The horizontal asymptote is y=3
 * x || 10 || 100 || 1000 || 10000 || 100000 || 1000000 ||
 * F(x) || 3.2 || 3.1 || 3.01 || 3.001 || 3.0001 || 3.00001 ||