12.4 Limits at Infinity and Limits of Sequences

Limits at Infinity

Consider the Limits:



The limit is looking for value of as gets arbitrarily close to .




The limit is looking for the value of as gets arbitrarily close to .

If we plug in very large and super small numbers for , such as 1 billion and -1 billion, the results come out to about in each case.

Another Example:



and



Again, if we plug in numbers representing the far left and right ends of the graph, we find that in this case the limit Does Not Exist because the limits are different at either end.

A third example:



and



If we plug in our large and small numbers again, we notice that on both ends, the graph approaches .

Does any of this seem familiar?

• the value being the ratio of the coefficients of the leading terms when the degrees of the leading terms are EQUAL?
• no solution when the degree of the leading term of the numerator is HIGHER than the degree of the leading term of the denominator?
• the value being when the degree of the leading term of the numerator is LOWER than the degree of the leading term of the denominator?

It should: For finding horizontal asymptotes, the system is the same!!


Note:




For example:

Piecewise Functions

• Absolute Value Functions
  

• Inverse Tangent Functions

Limits of Sequences

Because a sequence is defined as being a function with a domain of natural numbers, the limits of sequences are the same as limits of functions: