a) (i) Define a function and give an example (ii) Given that f(x)=(6-4x)/(16-5x) and g(x)=(-3x)/(1-4x) Find g_(0)f at x=1 (3 marks) b) Evaluate the following limits (i) lim _(xarrow 0)((sqrt (1+x)-sqrt (1-x))/(x)) (3 marks) (ii) lim _(xarrow infty )((2x^3-4x+6)/(3x^2)-x+20) (2 marks) c) Use the first principle to find the rate of change of y with respect to x given y(x)=(4-5x)/(3-2x) (1 mark)

a) (i) A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output. An example of a function is , where for each input value of , there is a unique output value of .(ii) To find at , we need to evaluate . First, we find by substituting into the function : Next, we substitute this value into the function : b) (i) To evaluate the limit , we can use the conjugate method. Multiply the numerator and denominator by the conjugate of the numerator: (ii) To evaluate the limit , we can divide the numerator and denominator by the highest power of in the denominator: c) To find the rate of change of with respect to using the first principle, we need to find the derivative of with respect to . The first principle states that the derivative of a function at a point is given by: In this case, . Let's find the derivative using the first principle: Simplifying the expression and taking the limit as approaches 0, we get: After simplifying further, we get: Taking the limit as approaches 0, we get: