The Indeterminate Form Some limits can be found by using direct substitution. When using direct substitution, if the result is a finite number, then that number is the limit of the function. If the result is in the form , this is an Indeterminate Form. In this case, one of the following techniques can be used: Factor the expression, simplify, then use direct substitution. Rationalize the numerator, simplify, then use direct substitution. Simplify the complex fraction, then use direct substitution.

Factoring Example Direct substitution yields an indeterminate form, so we must factor. Find the limit of: Factor Simplify Substitute

Rationalizing Example Direct substitution yields an indeterminate form, so we must rationalize. Find the limit of: Rationalize Simplify Substitute

Complex Fraction Example Direct substitution yields an indeterminate form, so we must simplify the complex fraction. Find the limit of: Simplify Substitute

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Special Trigonometric Limits Let’s see what the graph looks like. Now let’s look at this graph. As x approaches 0, f(x) approaches 1 As x approaches 0, f(x) approaches 0 That was easy

Different Formats of the Special Trigonometric Limits Let’s look at those special limits one more time. The symbol x can be replaced with any other expression as long as that expression also approaches 0. As x approaches 0, 3x also approaches 0. Factor out Multiply both sides of the equation by 3.

Trigonometric Limit Examples That’s just the upside-down version of the first one. Hey, that looks familiar. I can do this. We know that If we factor, we can use this to solve our problem. Multiply both sides by 4. That was easy

The Squeeze Theorem If for all x in an open interval containing c, except possibly at c itself, and if then Don’t worry, that just means if f(x) is between h(x) and g(x), and the limit of h(x) is equal to the limit of g(x), then the limit of f(x) is also that same limit. g f h c

Squeeze Theorem Example Use a graphing calculator to graph each function and observe that the squeeze theorem exists. Then find the limit as x approaches 0 analytically for each function.

Another Squeeze Theorem Example Use the Squeeze Theorem to find

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