Question

I wanted to know if the statement is true or false. If it is true explain why. If it is false explain why.
a) if f(x) = x^3 then lim x->3(f(x)-f(3))/(x-3) =27
b) d/dx(cos pi/4) = -sin (pi/4)

Answer 1

a) lim x->3 (f(x)-f(3))/(x-3) =27, f(x) = x^3
lim x->3 (x^3)-3^2/(x-3)
now factor the numerator.
The factors of the difference of two cubes are given by

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

If you are interested in learning something instead of having your homework done for you, stop now and see if you can solve the limit with the above fact from basic algebra.

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lim x->3 (x^3)-3^2/(x-3) = lim x->3 [(x-3)(x^2 + 3x + 9)]/(x-3)
= lim x->3 (x^2 + 3x + 9)
now there is no danger of division by zero so the limit can be solved by substitution
lim x->3 (x^2 + 3x + 9)= 3^2 + 3(3) + 9 = 27

b) is true, but I'm not going to prove the basic d/dx cos(x) = -sin(x) here.