Infinity Minus Infinity

Often, particularly with fractions, l'Hôpital's Rule can help in cases where one term with infinite limit is subtracted from another term with infinite limit. This is generally done by finding common denominators. For example:

the limit as x goes to 1 of (x/(x-1)) - (1/(ln(x))) = the limit as x goes to 1 of (x*ln(x)-x+1)/((x-1)*ln(x)).

Now, we have just one fraction, and both the numerator and denominator have a limit of zero. So, we can apply l'Hôpital's Rule.

the limit as x goes to 1 of (x*ln(x)-x+1)/((x-1)*ln(x)) *= the limit as x goes to 1 of (1*ln(x)+(x*(1/x))-1)/(1*ln(x)+(x-1)/x) = the limit as x goes to 1 of (ln(x)+1-1)/(ln(x)+(x-1)/x) =

the limit as x goes to 1 of (ln(x))/((x*ln(x)+(x-1))/x) = the limit as x goes to 1 of (x*ln(x))/(x*ln(x)+(x-1))

We can now use l'Hôpital's Rule again, since the limits of the numerator and denominator are again zero, using the Product Rule carefully on the top and bottom of the fraction.

the limit as x goes to 1 of (x*ln(x))/(x*ln(x)+(x-1)) *= the limit as x goes to 1 of (1*ln(x)+(x*(1/x)))/(1*ln(x)+(x*(1/x))+1) =

the limit as x goes to 1 of (ln(x)+1)/(ln(x)+2) = (ln(1)+1)/(ln(1)+2) = (0+1)/(0+2) = 1/2

Copyright © 1996 Department of Mathematics, Oregon State University

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