Flashcards in 2.2.3 Two Techniques for Evaluating Limits Deck (7):
Two Techniques for Evaluating Limits
• When evaluating the limit of a compound fraction, try to simplify the fraction by finding the lowest common denominator.
• An expression involving a binomial can often be simplified by multiplying by the conjugate of the binomial. Given a binomial expression ( a + b ), the conjugate is the expression ( a – b ).
- Attempting direct substitution with this limit results in an indeterminate form.
- This expression is a compound fraction; it has a fraction in the numerator. You will need to simplify the numerator by finding a common denominator.
- Dividing by a fraction is accomplished by multiplying by its reciprocal. Cancellation removes the 0/0 culprit.
- Now direct substitution produces the value of the limit.
- Here is another limit where direct substitution results in an indeterminate form. However, there is nothing to factor and nothing to combine.
- You can remove the radical from the numerator if you multiply by its conjugate. The numerator is of the form a – b, so multiply the numerator and denominator by a + b. In this way you are essentially multiplying by 1.
- Notice that the radical is now in the denominator. You may not think that you have made any progress, but now you can cancel the factors of x in the numerator and denominator.
- Direct substitution then produces the value of the limit.
Evaluate the limit lim x→2 x−√5x−6/x^2−4.
Evaluate the limit lim h→0 1/(1+h)^2−1/h.
Evaluate the limit lim x→0 x/√x+4−2.
Evaluate the limit lim x→2 1/x−1/2 / x−2.