## 3.6.8 Finding Limits

In doing many kinds of calculations, you need to evaluate expressions when variables take on particular values. In many cases, you can do this simply by applying transformation rules for the variables using the /. operator.

In some cases, however, you have to be more careful.
Consider, for example, finding the value of the expression when . If you simply replace by in this expression, you get the indeterminate result . To find the correct value of when , you need to take the limit.

Finding limits.

Not all functions have definite limits at particular points. For example, the function oscillates infinitely often near , so it has no definite limit there. Nevertheless, at least so long as remains real, the values of the function near always lie between and 1. Limit represents values with bounded variation using Interval objects. In general, Interval[xmin, xmax] represents an uncertain value which lies somewhere in the interval xmin to xmax.

Some functions may have different limits at particular points, depending on the direction from which you approach those points. You can use the Direction option for Limit to specify the direction you want.

Directional limits.

Limit makes no assumptions about functions like f[x] about which it does not have definite knowledge. As a result, Limit remains unevaluated in most cases involving symbolic functions.