How do you evaluate the limit #cos((x^5+1)/(x^6+x^5+100))# as x approaches #-oo#?

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Answer 1 · 2016-08-30T17:04:56

The Reqd. Limit#=1#.

The Reqd. Limit #=lim_(xrarr-oo) cos {(x^5+1)/(x^6+x^5+100)}#

#=lim_(xrarr-oo) cos {(x^5(1+1/x^5))/(x^6(1+1/x+100/x^6))}#

#=lim_(xrarr-oo) cos {(1/x)((1+1/x^5)/(1+1/x+100/x^6))}#

Since #cos# is continuous on #RR#, we have,

The Reqd. Limit#=cos {lim_(xrarr-oo) (1/x)((1+1/x^5)/(1+1/x+100/x^6))}#

#=cos {0*((1+0)/(1+0+0))}#.

#=1#.