Lim x-> infinity (1-(e^x))/(1+(2e(^x)))
EXPERT 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