Q. 12 - Calculus

Question

Explain how the algebraic function $f\left(x\right)=\frac{\left(x^2+1\right)\left(4-3x\right)}{3x^2}$ is a combination of identity, constant, and power functions. Why does this mean that we can calculate limits of this function at domain points by evaluation? 

Step-by-Step Solution

Answer

By the sum, product and division rules for limits, and the fact that power and constant functions are continuous, we know that this function f  is continuous and thus we can calculate its limits at domain points by evaluation. 

Step 1. Given information

The given function $f\left(x\right)=\frac{\left(x^2+1\right)\left(4-3x\right)}{3x^2}$

Step 2. The strategy is to explain that the above function is a combination of identity, constant and power functions.

To get this function we add the power function $x^2$ and the constant term 1 , and then multiply the result with another function which is the difference of function 4 and constant multiple function $3x$ . Now divide the result by the power function x² which is again a constant multiple function, multiplied by the constant 3 . Thus by the sum, product and division rules for limits, and the fact that power and constant functions are continuous, we know that this function f  is continuous and thus we can calculate its limits at domain points by evaluation.