2.2.3 Two Techniques for Evaluating Limits

Question 1

Two Techniques for Evaluating Limits

Answer 1

• 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 ).

Question 2

note

Answer 2

• 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.

Question 3

Evaluate the limit lim x→2 x−√5x−6/x^2−4.

Answer 3

−1/16

Question 4

Evaluate the limit lim h→0 1/(1+h)^2−1/h.

Answer 4

-2

Question 5

Evaluate the limit lim x→0 x/√x+4−2.

Answer 5

4

Question 6

Evaluate the limit lim x→2 1/x−1/2 / x−2.

Answer 6

-1/4

Question 7

Evaluate the limit lim x→1 [1/x−1 − 2/x^2−1].

HINT: Remember to combine the two fractions using a common denominator before evaluating the limit.

Answer 7

1/2