Evaluate the following limit. If the answer is positive infinite, type "I"; if negative infinite, type "N"; and if it does not exist, type "D". lim x->infinity tan^-1(x^2-x^4)?

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Answer 1 · 2018-04-17T02:32:35

#-pi/2#

#lim_(x->oo)tan^-1(x^2-x^4)=tan^-1(lim_(x->oo)x^2-x^4)#

We can move the limit inside as the arctangent is a continuous function. Evaluating, we obtain

#tan^-1(lim_(x->oo)x^2-x^4)=tan^-1(oo-oo)#

This is an indeterminate form and does not tell us anything. We need to simplify.

#tan^-1(lim_(x->oo)x^2-x^4)=tan^-1(lim_(x->oo)x^2(1-x^2))#

We may evaluate this, yielding

#tan^-1(lim_(x->oo)x^2(1-x^2))=tan^-1(oo(-oo))=tan^-1(-oo)=-pi/2#

#oo(-oo)=-oo,# as multiplying a very large positive number by a very large negative number yields a very large negative number.

Furthermore, we know the following basic limits for the arctangent:

#lim_(x->oo)tan^-1(x)=pi/2, lim_(x->oo)tan^-1(x)=-pi/2#