Question

limit as delta x approaches 0 when f(x) = 4/ sqr x


Have to use the (F(x+delta x)-f(x))/delta x method

Answer 1

just plug and chug

[f(x+h)-f(x)]/h

4(x+h)^(-1/2) - 4*x^(-1/2)

Now, to evaluate (x+h)^(-1/2), use the binomial theorem:
(a+b)^n = a^n + n*a^(n-1)*b^1 + n(n-1)/2 * a^(n-1)b^2 + ...

(x+h)^(-1/2) = x^(-1/2) + (-1/2)*x^(-3/2)*h + (-1/2)(-3/2)/2 * x^(-5/2)h^2 + ...

(x+h)^(-1/2) - x^(-1/2) = (-1/2)*x^(-3/2)*h + (-1/2)(-3/2)/2 * x^(-5/2)h^2 + ...

divide all that by h to get

(-1/2)*X^(-3/2) + (-1/2)(-3/2)/2 * x^(-5/2)h + ...

= -1/2x^(-3/2) + 3/8x^(-5/2)*h + ...

As h->0, all the terms with h disappear, leaving just

-1/2 x^(-3/2)

multiply that by 4 from the original function, and you get

-2x^-3/2 = -2/√x³