Limits | Engineering Mathematics and Sciences

\begin{align*} \displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h} & =\frac{\left[\left(x+h\right)^2+1\right]-\left(x^2+1\right)\:}{h}\\ \\ & =\frac{x^2+2xh+h^2+1-x^2-1}{h}\\ \\ & =\frac{2xh+h^2}{h}\\ \\ & =\frac{h\left(2x+h\right)}{h}\\ \\ & =2x+h \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\ \end{align*}

If $\displaystyle f\left(x\right)=x^2-4x$ , find

a) $\displaystyle f\left(-5\right)$

b) $\displaystyle f\left(y^2+1\right)$

c) $\displaystyle f\left(x+\Delta x\right)$

d) $\displaystyle f\left(x+1\right)-f\left(x-1\right)$

Part A

\begin{align*} f\left(-5\right) & =\left(-5\right)^2-4\left(-5\right)\\ & =25+20\\ & =45 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B

\begin{align*} f\left(y^2+1\right) & = \left(y^2+1\right)^2-4\left(y^2+1\right)\\ & =y^4+2y^2+1-4y^2-4\\ & =y^4-2y^2-3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part C

\begin{align*} f\left(x+\Delta x\right)&=\left(x+\Delta x\right)^2-4\left(x+\Delta x\right)\\ & =\left(x+\Delta x\right)\left[\left(x+\Delta x\right)-4\right]\\ & =\left(x+\Delta x\right)\left(x+\Delta x-4\right) \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

Part D

\begin{align*} f\left(x+1\right)-f\left(x-1\right) & =\left[\left(x+1\right)^2-4\left(x+1\right)\right]-\left[\left(x-1\right)^2-4\left(x-1\right)\right]\\ & = \left[x^2+2x+1-4x-4\right]-\left[x^2-2x+1-4x+4\right]\\ & =x^2-x^2+2x-4x+2x+4x+1-4-1-4\\ & =4x-8\\ & =4\left(x-2\right) \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 8

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 1

If $\displaystyle f\left(x\right)=x^2-4x$ , find

a) $\displaystyle f\left(-5\right)$

b) $\displaystyle f\left(y^2+1\right)$

c) $\displaystyle f\left(x+\Delta x\right)$

d) $\displaystyle f\left(x+1\right)-f\left(x-1\right)$

Part A

Part B

Part C

Part D

If \displaystyle f\left(x\right)=x^2+1, find \displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0.

If \displaystyle f\left(x\right)=x^2-4x, find

a) \displaystyle f\left(-5\right)

b) \displaystyle f\left(y^2+1\right)

c) \displaystyle f\left(x+\Delta x\right)

d) \displaystyle f\left(x+1\right)-f\left(x-1\right)

Part A

Part B

Part C

Part D

If $\displaystyle f\left(x\right)=x^2+1$ , find $\displaystyle \frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne 0$ .

If $\displaystyle f\left(x\right)=x^2-4x$ , find

a) $\displaystyle f\left(-5\right)$

b) $\displaystyle f\left(y^2+1\right)$

c) $\displaystyle f\left(x+\Delta x\right)$

d) $\displaystyle f\left(x+1\right)-f\left(x-1\right)$