How do you find horizontal asymptotes using limits?

Question

Elijah Abram · Accepted Answer

Evaluate the limits as #x# increases without bound (#xrarroo#) and as #x# decreases without bound (#xrarr-oo#).

For function, #f#, if #lim_(xrarroo)f(x) = L# (That is, if the limit exists and is equal to the number, #L#), then the line #y=L# is an asymptote on the right for the graph of #f#. (If the limit fails to exist, then there is no horizontal asymptote on the right.) if #lim_(xrarr-oo)f(x) = L# (That is, if the limit exists and is equal to the number, #L#), then the line #y=L# is an asymptote on the left for the graph of #f#. (If the limit fails to exist, then there is no horizontal asymptote on the left.) For rational functions, if one of the limits at infinity exists, then the other does as well and they are equal. Other types of functions may have an asymptot on one side, but not the other -- e.g #f(x) = e^x# or may have different left and right horizontal asymptotes -- e.g. #f(x) = arctan x#

Isaiah Allen · Answer

undefined To find horizontal asymptotes using limits, you need to evaluate the limit of the function as x approaches positive or negative infinity.

1. If the limit is a finite number, then there is a horizontal asymptote at that value.
2. If the limit is positive or negative infinity, then there is no horizontal asymptote.
3. If the limit does not exist, then there is no horizontal asymptote.

To find the limit, you can simplify the function and analyze the highest power of x in the numerator and denominator.

- If the highest power of x in the numerator is less than the highest power of x in the denominator, the horizontal asymptote is y = 0.
- If the highest power of x in the numerator is equal to the highest power of x in the denominator, divide the coefficients of the highest power terms to find the horizontal asymptote.
- If the highest power of x in the numerator is greater than the highest power of x in the denominator, there is no horizontal asymptote.

Remember to consider any restrictions on the domain of the function and to analyze the behavior of the function as x approaches positive or negative infinity.