Question

A. Use properties of limits and algebraic methods to find the limit, if it exists. (If the limit is infinite, enter '[infinity]' or '-[infinity]', as appropriate. If the limit does not otherwise exist, enter DNE.)

lim (x→−1/2) = (4x − 2)/(8x2 + 1)

b. Differentiate the function g(x) = 1/(2x^3 + 5x + 4)^(3/4)

Answer 1

a. -4/3

b. -3(6x^2 +5)/(4(2x^3 +5x +4)^(7/4))

Explanation:

a.

The function can be evaluated at x = -1/2:

(4(-1/2) -2)/(8(-1/2)^2 +1) = (-2 -2)/(8/4 +1) = -4/3

The limit at x = -1/2 is -4/3.

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b.

The power rule and chain rule will get you there:

d(u^n) = nu^(n-1)·du

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g(x) = (2x^3 +5x +4)^(-3/4)

g'(x) = (-3/4)(2x^3 +5x +4)^(-7/4)(6x^2 +5)

or ...

g'(x) = -3(6x^2 +5)/(4(2x^3 +5x +4)^(7/4))