Calculus

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Work out the problem and explain your steps as you go please!

1. Find the limit by numerical evidence of limx -->4 x3+4x2-7x-5

2. Find the limit of limx-->-9 x2-81x+9.

3. Find the limit of each of the following:
a. limx-->0 x+4x-2x

b. limx-->0 x3+51 x2-17

c. limn-->infinity 1+1nn

d. limx-->0 sinxx

  • Calculus -

    #1.
    f(3.000000) = 37.000000
    f(3.900000) = 87.859000
    f(3.990000) = 94.271599
    f(3.999000) = 94.927016
    f(3.999900) = 94.992700
    f(3.999990) = 94.999270
    f(3.999999) = 94.999927
    Looks like lim = 95, eh?

    do #2 likewise

    for lim sin(x)/x use L'Hopital's Rule

    lim sinx/x = lim cosx/1 = 1
    ---------------------

    lim [1 + (1/x)]^x ...... it is understood that x approaches infinity
    = lim e^ln( [1 + (1/x)]^x )
    = e^ ( lim ln( [1 + (1/x)]^x ))
    = e^ ( lim x * ln[1 + (1/x)] )
    = e^ ( lim ln[1 + (1/x)] / (1/x) ) .. the limit is now 0/0 ...
    ..... by L'Hopitals Rule

    = e^ ( lim {1/[1 + 1/x] *(-1/x^2)} / {-1/x^2} )
    = e^ lim 1/[1+1/x]
    = e^1
    = e.

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