What is # lim_(->-oo) f(x) = sinx/(x^2-8)-1/(9-x^2)#?

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Answer 1 · 2015-12-01T16:34:19

# lim_(x->-oo) (sinx/(x^2-8)-1/(9-x^2)) = 0#

I suggest beginning by observing that

#sinx/(x^2-8)-1/(9-x^2) = sinx/(x^2-8) + 1/(x^2-9)#.

Now, consider each limit separately.

Use the squeeze theorem (at infinity) to show that

#lim_(xrarr-oo)sinx/(x^2-8) = 0#.

Use limits of rational functions to see that

#lim_(xrarr-oo)1/(x^2-9) = 0#.

Finally use properties of limits to conclude that the limit of the sum is the sum of the limits, so

# lim_(x->-oo) (sinx/(x^2-8)-1/(9-x^2)) = lim_(x->-oo)(sinx/(x^2-8) + 1/(x^2-9)) = 0#