Q. B - Calculus

Question

Limits that define derivatives: In the next chapter we will be interested in derivatives, which we will define as limits of the form

$\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h}$

(a) Calculate this limit for $f\left(x\right)=x^3$and $c=0.$

(b) Calculate this limit for $f\left(x\right)=x^3$and $c=2.$

(c) Calculate this limit for $f\left(x\right)=x^3$and general $c=x.$This time your answer will be a function of x instead of a number.

Step-by-Step Solution

Answer

(a) limit for $f\left(x\right)=x^3 \& c=0$is $0.$

(b) limit for $f\left(x\right)=x^3 \& c=2$is $12.$

(c) limit for $f\left(x\right)=x^3 \& c=x$is $3x^2.$

Part (a) Step 1. Given information.

The given limit that defines derivatives is $L=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h}.$

The given function is $f\left(x\right)=x^3.$

$c=0$

Part (a) Step 2. Limit for function.

The limit for the function $f\left(x\right)=x^3 \& c=0$is as follows.

$$\begin{gathered} L=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h} \\ L=\underset{h\to 0}{\lim }\frac{f(0+h)-f\left(0\right)}{h} \\ L=\underset{h\to 0}{\lim }\frac{{\left(h\right)}^3-{\left(0\right)}^3}{h} \\ L=0 \end{gathered}$$

So limit is $L=0.$

Part (b) Step 1. Given information.

The given limit that defines derivatives is $L=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h}.$

The given function is $f\left(x\right)=x^3.$

The given value of c is  $c=2.$

Part (b) Step 2. Limit for function.

The limit for the function $f\left(x\right)=x^3 \& c=2$is as follows.

$$\begin{gathered} L=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h} \\ L=\underset{h\to 0}{\lim }\frac{f(2+h)-f\left(2\right)}{h} \\ L=\underset{h\to 0}{\lim }\frac{{(2+h)}^3-{\left(2\right)}^3}{h} \\ L=\underset{h\to 0}{\lim }\frac{h^3+8+6h^2+12h-8}{h} \\ L=\underset{h\to 0}{\lim }{h}^2+6h+12 \\ L=12 \end{gathered}$$

So limit is $L=12.$

Part (a) Step 1. Given information.

The given limit that defines derivatives is $L=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h}.$

The given function is $f\left(x\right)=x^3.$

The given value of c is $c=x.$

Part (c) Step 2. Limit for function.

The limit for the function $f\left(x\right)=x^3 \& c=x$is as follows.

$$\begin{gathered} L=\underset{h\to 0}{\lim }\frac{f(c+h)-f\left(c\right)}{h} \\ L=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h} \\ L=\underset{h\to 0}{\lim }\frac{{(x+h)}^3-{\left(x\right)}^3}{h} \\ L=\underset{h\to 0}{\lim }\frac{h^3+x^3+3h^{2}x+3x^{2}h-x^3}{h} \\ L=\underset{h\to 0}{\lim }{h}^2+3hx+3x^2 \\ L=3x^2 \end{gathered}$$

So limit is $L=3x^2.$