Q. 11 - Calculus

Question

Explain how the algebraic function $f\left(x\right)={\left(\sqrt{x}+1\right)}^3$ 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 and composition 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 algebraic function $f\left(x\right)={\left(\sqrt{x}+1\right)}^3$

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 $\sqrt{x}$ and the constant term 1 , and then compose the result with the power function $x^3$. Thus by the sum and composition 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.