evaluate limit y --> ln(2) e^3y - 8 --------- e^2y - 4 Solution lim(y->ln(2))[(e

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lim(y->ln(2))[(e^(3y) - 8) / (e^(2y) - 4)] = lim(y->ln(2))[(e^y)^3 - 2^3) / ((e^y)^2 - 2^2)] = lim(y->ln(2))[(e^y - 2)(e^(2y) + 2e^y + 4) / ((e^y - 2)(e^y + 2)] = lim(y->ln(2))[(e^(2y) + 2e^y + 4) / (e^y + 2)] = (4 + 4 + 4)/(2 + 2) = 12/4 = 3. Note that in going from line 2 to line 3 I used the standard difference of cubes and difference of squares formulas. Note also, for example, that with y = ln(2), e^(2y) = e^(2ln(2)) = e^(ln(2^2)) = e^(ln(4)) = 4.

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