4 Algebraic Limits So far Numerically Graphically What
4. Algebraic Limits
So far …. . • Numerically • Graphically • What if you don’t have a graph or a calculator or your brilliant friend sitting next to you? • We will look at various algebraic methods. Their names are not important, but it is important to recognize their forms.
1. Direct Substitution • This is the method you should ALWAYS try first! • Example 1
2. VA • If you use direct substitution and get , that means there is a VA and the answer will be DNE or if it is a one -sided limit • Example 2
Indeterminate form • We say that f(x) is indeterminate at x = c if when we evaluate f(c), we obtain an undefined expression • Strategy is to transform f(x) algebraically into a new expression that is defined and continuous at x=c. • Remember that if you get something like NOT indeterminate then this is
3. Factor and divide out • The factor that causes 0/0 in the denominator can be simplified with its common factor in the numerator and then direct substitution can be used on the resulting function • Example 3
4. Rationalization conjugation • This method works well where there is a sum or difference of one or more radical terms. It involves multiplying by conjugate/conjugate • Example 4
5. Least common denominator • This method works well if there is a compound fraction or a complex fraction (a fraction within a fraction. ) It involves finding the LCD • Example 5
6. Expand • This method works well if there is some obvious math to do like expanding binomials • Example 6
7. Trig Manipulation • We have already seen this method, it involves using trig identities to rewrite limits to be ones we have memorized • Example 7
8. General cleverness • Try this when all else fails. • Example 8
summary Real # Compare Exponents (this is your answer) VA Plug in # to left and/or right is answer or other indeterminant Algebra to simplify Then plug in
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