obviously we have a problem. the top and bottom go to zero. since we have a 0/0 as we take the limit, we can use L'Hoptial's rule (do you remember it?)

to make life easier, i will also use a limit theorem here that says lim A*B = (lim A)*(lim B)

so lim t->0 of (t^(3))/[tan(2t)]^(3)

Apply L'hopital's, we get

lim t->0 of (3t^2)/[3(tan(2t))^2 * 2sec^2(2t)]

we still get 0/0 as we take the limit, we will apply l'hopital's again. but before i do, i want to split up this function, so i don't have to find the derivative of [3(tan(2t))^2 * 2sec^2(2t)]

notice that (3t^2)/[3(tan(2t))^2 * 2sec^2(2t)] = (3t^2)/[3(tan(2t))^2] *1/( 2sec^2(2t))

so lim t->0 of (3t^2)/[3(tan(2t))^2 * 2sec^2(2t)]

= lim t->0 of (3t^2)/[3(tan(2t))^2] * 1/(2sec^2(2t))

= lim t->0 of (3t^2)/[3(tan(2t))^2] * lim t->0 1/(2sec^2(2t))

now the second limit is okay, we will do l'hopital's on the first, we get:

= lim t->0 of (6t)/[6(tan(2t)) * 2sec^2(2t)] * lim t->0 1/(2sec^2(2t))

we still have a problem with the first limit. i split it again and perform l'hopital's on the first piece

lim t->0 of (6t)/[6(tan(2t)) * 2sec^2(2t)] * lim t->0 1/(2sec^2(2t))

= lim t->0 of (6t)/[6(tan(2t))] * lim t->0 1/(2sec^2(2t)) * lim t->0 1/(2sec^2(2t))

apply l'hopital's to the first:

= lim t->0 of (6)/[6*2sec^2(2t)] * lim t->0 1/(2sec^2(2t)) * lim t->0 1/(2sec^2(2t))

now we can plug in. notice that sec^2(2t) = 1/cos^2(2t) and cos(0) = 1

taking the limits we obtain:

= 1/2 * 1/2 * 1/2

= 1/8

the first term is ok for taking limits, but the second piece, sqrt(1+sin(t))]/(t^(3)) has a problem. the limit makes the denominator zero, and we can't have that. however, we can't use l'hoptial's on this one since we don't have 0/0 or infinity/infinity, and thank God for that, since differentiating that 3 times over will be a pain. so what do we do.lim t->0 of [sqrt(1+tan(t)) - sqrt(1+sin(t))]/(t^(3))

notice that we have 1/0 as we take the limit, so it would hint that the limit for this piece is infinity, but not so fast, we have to make sure of that. if we approach zero from the left the limit is -infinity, if we approach it from the right, the limit is +infinity (do you see that?). since the right hand limit is not equal to the left hand limit, the limit does not exist. so the limit of this whole functiondoes not exist.

y = x^3 - 2x^2 + 1evaluate dy if y=x^(3) - 2x^(2) + 1 if x=2 and dx= 0.2

=> dy/dx = 3x^2 - 4x

=> dy = dx(3x^2 - 4x) ............i multiplied through by dx

when dx = 0.2 and x = 2 we get

dy = (0.2)(3(2)^2 - 4(2)

=> dy = 0.8

we will use linear approximation using differentials to do this problem.approximate sqrt(64.05) using differentials.

recall that f(x) ~= f(a) + f ' (a)(x - a)

x is what we want to know, a is something close to x that we already know. Huh??? let me explain.

we want to know sqrt(64.05). if we write this as a function we can write it as f(x) = sqrt(x), and we want to know f(64.05).

so what is f(64.05). i dunno, but i sure as heck know what f(64) is. sqrt(64) = 8. and 64 is close to 64.05.

so now we use x = 64.05 and a = 64, let's do the problem.

Consider the function f(x) = sqrt(x)

now f ' (x) = 1/2*sqrt(x)

using f(x) ~= f(a) + f ' (a)(x - a) with x = 64.05 and a = 64, we will approximate the value of f(64.05), that is sqrt(64.05)

Now f(64.05) ~= f(64) + f ' (64)(64.05 - 64)

=> f(64.05) ~= sqrt(64) + 1/2sqrt(64) * (0.05)

=> f(64.05) ~= 8 + 0.0625 * (0.05)

=> f(64.05) ~= 8.003125

so sqrt(64.05) ~= 8.003125 by the method of differentials.

in case you curious, the actual value is 8.00312439, pretty close huh!