Math Questions and Answers

Start Your Free Trial

Find the following limits, using L'Hospital's rule, if appropriate. Use INF to denote infinity and MINF to denote negative infinity lim as x approaches infinity of ((tan^(-1))(x/3))/((sin^(-1))(1/x))    

Expert Answers info

Luca B. eNotes educator | Certified Educator

calendarEducator since 2011

write5,348 answers

starTop subjects are Math, Science, and Business

You need to substitute `oo ` for x in equation under limit such that:

`lim_(x-gtoo) (tan^(-1)(x/3))/(sin^(-1)(1/x)) = oo/oo`

You need to use l'Hospital's theorem such that:

`lim_(x-gtoo) ((tan^(-1)(x/3))')/((sin^(-1)(1/x))') = lim_(x-gtoo) (1/(3(1 + x^2/9)))/(-(1/(x^2sqrt(1 - x^2/9))))`

`lim_(x-gtoo) (1/(3(1 + x^2/9)))/(-(1/(x^2sqrt(1 - x^2/9)))) = -(1/3)lim_(x-gtoo) (x^2sqrt(1 - x^2/9))/(x^2(1/x^2 + 1/9))`

`-(1/3)lim_(x-gtoo) (x^2sqrt(1 - x^2/9))/(x^2(1/x^2 + 1/9)) = -(1/3)lim_(x-gtoo) (sqrt(1 - x^2/9))/((1/x^2 + 1/9)) = oo`

Hence, evaluating the limit to the function yields `lim_(x-gtoo) (tan^(-1)(x/3))/(sin^(-1)(1/x)) = oo.`

check Approved by eNotes Editorial