Use algebraic manipulations to evaluate the limit below.
a) lim x -> infinity, 4e^x + 8 / 11e^x + 9
b) lim x -> infinity, x + 9 / 10 - x
Steps preferred. Thanks in advance.
a) lim x -> infinity, 4e^x + 8 / 11e^x + 9
b) lim x -> infinity, x + 9 / 10 - x
Steps preferred. Thanks in advance.
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I believe in both cases all you need is to make an approximation. As e^x is a far more significant factor than the rest ( + 8, - 9, etc), then you can essentially ignore those (as x approaches infinity the e^x terms are going to be far bigger that those, in other words).
So for instance in a), you can treat it as
lim x-> infinity, 4e^x/11e^x
Then just divide by e^x on the top and bottom to get 4/11.
As for b), it's the same deal: approximate to x/-x, and cancel the x's to get -1 (as you can see, if x is say 1000, 1009/-995 is already close to -1, and with x = a million, 1,000,009/-999,995 is closer still to -1).
So for instance in a), you can treat it as
lim x-> infinity, 4e^x/11e^x
Then just divide by e^x on the top and bottom to get 4/11.
As for b), it's the same deal: approximate to x/-x, and cancel the x's to get -1 (as you can see, if x is say 1000, 1009/-995 is already close to -1, and with x = a million, 1,000,009/-999,995 is closer still to -1).