Calculus I, Chapter 3: The Average Rate of Change and Limits
Previous Section: 3.7 Evaluating Limits with Algebra
For some limit problems, the methods from the previous section aren’t enough. This section will cover a couple of unique types of problems pertaining to trigonometry, substitution and distributing limits in composite functions.
Trigonometric Limits
When evaluating limits with trigonometric functions, the easiest method can involve rewriting each part in terms of sin(x) and cos(x). Otherwise, finding ways to use double-angle, square, and other trigonometric formulas to your advantage will be necessary. Because of the variety of methods and the lack of consistency, these types of limits can be pretty difficult to solve.
Example 1:
Substitution
As a result of continuity, we are able to evaluate limits by substitution if they are continuous or have a known answer. This is pretty straightforward. If it is continuous it is the case that a function both exists and has limits, and they are equal. A continuous function is fairly easy to identify as it will not have any discontinuities which occur primarily at domain restrictions. So, if the function exists at the point we are evaluating there is a pretty good chance that the limit can be solved using substitution.
Example 4:
Composite Limits
When solving a limit of the following form lim┬(x→a)〖f(g(x))〗 if the function f(x) doesn’t contain any extra variables other than g(x) itself, then the limit can be distributed to g(x), as in f( lim┬(x→a) g(x) ). Examples of this include trigonometric functions, exponential functions, and logarithmic functions.
If the resulting limit inside of the composition does not have an answer, it is not necessarily the case that the overall limit does not have an answer. However, if the limit does exist, then the limit can be evaluated by substitution.
Example 5:
Review
To solve trigonometric limits, first try to use sine and cosine, then use other trigonometric formulas to simplify if needed
Continuous functions can be evaluated by substituting
In composite functions, a limit can be distributed to the interior functions