Infinity Times Zero

Consider

the limit as x goes to 0 from the right of ln(x)*sin(x).

In this case, there is no fraction in the limit. Since the limit of ln(x) is negative infinity, we cannot use the Multiplication Limit Law to find this limit. We can convert the product ln(x)*sin(x) into a fraction:

the limit as x goes to 0 from the right of ln(x)*sin(x) = the limit as x goes to 0 from the right of (ln(x))/(1/sin(x)) = the limit as x goes to 0 from the right of (ln(x))/(csc(x)).

Now, we have a fraction where the limits of both the numerator and denominator are infinite. Thus, we can apply l'Hôpital's Rule:

the limit as x goes to 0 from the right of (ln(x))/(csc(x)) *= the limit as x goes to 0 from the right of (1/x)/(-csc(x)*cot(x)) =

the limit as x goes to 0 from the right of (1/x)/(-(1/sin(x))(1/tan(x))) = the limit as x goes to 0 from the right of (-sin(x)*tan(x))/x.

Remember that the derivative of ln(x) is 1/x, and the derivative of csc(x) is -csc(x)cot(x).

We can now use l'Hôpital's Rule again, as the limits of both the top and the bottom are zero, using the Product Rule to take the derivative of the numerator.

the limit as x goes to 0 from the right of -(sin(x)*tan(x))/x *= the limit as x goes to 0 from the right of -(cos(x)*tan(x)+sin(x)*sec^2(x))/1 =

the limit as x goes to 0 from the right of -(sin(x)+(tan(x))/(cos(x))) = -(0+(0/1)) = 0

Copyright © 1996 Department of Mathematics, Oregon State University

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