On-line Math 21

3.1  Trigonometric functions

3.1.1  Limits of trig functions

Example 1

lim x® 0  cos(x)-1 x = 0

Solution

We could start over, going through the same sort of geometric-trigonometric arguments that we did for the limit of sin(x) . But, instead, we can be trickier.

cos(x)-1 x = ( cos(x)-1) ( cos(x)+1) x( cos(x)+1) = cos2(x)-1 x( cos(x)+1) = -sin2(x) x(cos(x)+1) .

Now, after that trick, the numerator has been transformed into sin(x) (well, two of them). But one of those sin(x) 's can be used to cover the x in the denominator:

lim x® 0  cos(x)-1 x = lim x® 0  -sin2(x) x(cos(x)+1) = - lim x® 0  æ ç è sin(x) x ö ÷ ø æ ç è sin(x) (cos(x)+1) ö ÷ ø = - æ ç è lim x® 0  -sin2(x) x(cos(x)+1) ö ÷ ø æ ç è lim x® 0  sin(x) (cos(x)+1) ö ÷ ø = -( 1) 0 2 = 0.

Copyright (c) 2000 by David L. Johnson.