How can i find the limit or show this exists. lim(x->infinity) (1-e^x)/(1+2e^x)

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Explanation / Answer

To find the lt --> infnity {(1-e^x)/(1+2e^x)}, we transform e^x = y and so when x-->ifinity, y = e^x -->ifinity. Therefore, Lt x--> ifinity {(1-e^x)/(1+2e^x)} = Lty-->infinity (1-y)/(1+2y) Lt x--> ifinity {(1-e^x)/(1+2e^x)} = Lty-->infinity (1-y)/(1+2y). We divide both numerator and denominator by y on the right: Lt x--> ifinity {(1-e^x)/(1+2e^x)} = Lty-->infinity (1/y-y/y)/(1/y+2y/y) Lt x--> ifinity {(1-e^x)/(1+2e^x)} = Lty-->infinity (1/y-1)/(1/y+2). Lt x--> ifinity {(1-e^x)/(1+2e^x)} = (0-1)/(0+2). Lt x--> ifinity {(1-e^x)/(1+2e^x)} = (0-1)/(0+2) = -1/2 = -0.5

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