Category Archives: Calculus

Includes differential, integral and series calculus

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 1

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 4}\left(\frac{x^3-64}{x^2-16}\right)$

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SOLUTION:

A straight substitution of $x=4$ leads to the indeterminate form $\frac{0}{0}$ which is meaningless.

Therefore, to evaluate the limit of the given function, we proceed as follows

\begin{align*} \lim\limits_{x\to 4}\left(\frac{x^3-64}{x^2-16}\right)& =\lim\limits_{x\to 4}\left(\frac{\left(x-4\right)\left(x^2+4x+16\right)}{\left(x+4\right)\left(x-4\right)}\right)\\ \\ & =\lim\limits_{x\to 4}\left(\frac{x^2+4x+16}{x+4}\right)\\ \\ & =\frac{\left(4\right)^2+4\left(4\right)+16}{4+4}\\ \\ & =\frac{48}{8}\\ \\ & =6 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 8

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 0}\left(\frac{3x+2}{x^2-2x+4}\right)$ .

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SOLUTION:

Plug in the value x=0.

\begin{align*} \lim\limits_{x\to 0}\left(\frac{3x+2}{x^2-2x+4}\right)& =\frac{3\left(0\right)+2}{\left(0\right)^2-2\left(0\right)+4}\\ \\ & =\frac{0+2}{0-0+4}\\ \\ & =\frac{2}{4}\\ \\ & =\frac{1}{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 7

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{\sqrt{3x}}{x\sqrt{x+1}}\right)$ .

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SOLUTION:

Plug the value x=3.

\begin{align*} \lim\limits_{x\to 3}\left(\frac{\sqrt{3x}}{x\sqrt{x+1}}\right)&=\frac{\sqrt{3\left(3\right)}}{3\sqrt{3+1}} \\ \\ &=\frac{\sqrt{9}}{3\sqrt{4}} \\ \\ & =\frac{3}{3\cdot 2}\\ \\ & =\frac{3}{6}\\ \\ &=\frac{1}{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 6

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PROBLEM:

Evaluate $\displaystyle \lim_{x\to 2}\left(4x-3\right)\left(x^2+5\right)$ .

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SOLUTION:

Plug the value x=2.

\begin{align*} \lim\limits_{x\to 2}\left(4x-3\right)\left(x^2+5\right) & =\left[\left(4\cdot 2\right)-3\right]\left[\left(2\right)^2+5\right]\\ & =\left[8-3\right]\left[4+5\right]\\ & =\left(5\right)\left(9\right)\\ & =45 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 5

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 8}\left(2x+\sqrt[3]{x}-4\right)$ .

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SOLUTION:

Plug in the value x=8.

\begin{align*} \lim\limits_{x\to 8}\left(2x+\sqrt[3]{x}-4\right) & = \left[2\left(8\right)+\sqrt[3]{8}-4\right]\\ & =\left[16+2-4\right]\\ & =14 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 4

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PROBLEM:

Evaluate $\displaystyle \lim\limits _{x\to \frac{\pi }{3}}\left(\frac{\sin\:2x}{\sin\:x}\right)$ .

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SOLUTION:

Plug in the value $\displaystyle x=\frac{\pi }{3}$ .

\begin{align*} \lim\limits_{x\to \frac{\pi }{3}}\left(\frac{\sin\:2x}{\sin\:x}\right) & =\frac{\sin\left(2\cdot \frac{\pi }{3}\right)}{\sin\:\left(\frac{\pi }{3}\right)} \\ & =\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2}}\\ & =1 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 3

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right)$ .

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SOLUTION:

\begin{align*} \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right) & =\lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x\right)+\lim\limits_{x\to \frac{\pi }{4}}\left(\sin\:x\right)\\ & =\tan\:\frac{\pi }{4}+\sin\:\frac{\pi }{4}\\ & =1+\frac{\sqrt{2}}{2}\\ & =\frac{2+\sqrt{2}}{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 2

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PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right)$ .

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SOLUTION:

\begin{align*} \lim_{x\to 3}\left(\frac{4x+2}{x+4}\right)& =\frac{\lim\limits_{x\to 3}\left(4x+2\right)}{\lim\limits_{x\to 3}\left(x+4\right)}\\ & =\frac{\lim\limits_{x\to 3}\left(4x\right)+\lim\limits_{x\to 3}\left(2\right)}{\lim\limits_{x\to 3}\left(x\right)+\lim\limits_{x\to 3}\left(4\right)}\\ & =\frac{4\cdot \lim\limits_{x\to 3}\left(x\right)+2}{3+4}\\ & =\frac{4\cdot 3+2}{3+4}\\ & =\frac{12+2}{7}\\ & =\frac{14}{7}\\ & =2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 1

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PROBLEM:

Evaluate $\displaystyle \lim _{x\to 2}\left(x^2-4x+3\right)$ .

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SOLUTION:

\begin{align*} \lim_{x\to 2}\left(x^2-4x+3\right)& = \lim_{x\to 2}\left(x^2\right)-\lim_{x\to 2}\left(4x\right)+\lim_{x\to 2}\left(3\right)\\ & =\left[\lim_{x\to 2}\left(x\right)\right]^2-4\lim_{x\to 2}\left(x\right)+3\\ & =\left(2\right)^2-4\left(2\right)+3\\ & =-1 \ \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 10

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PROBLEM:

If $\displaystyle f\left(x\right)=\frac{4}{x+3}$ and $\displaystyle \:g\left(x\right)=x^2-3$ , find $\displaystyle f\left[g\left(x\right)\right]$ and $\displaystyle g\left[f\left(x\right)\right]$ .

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SOLUTION:

Part A

\begin{align*} f\left[g\left(x\right)\right] & =\frac{4}{\left(x^2-3\right)+3}\\ & =\frac{4}{x^2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

Part B

\begin{align*} g\left[f\left(x\right)\right] & =\left(\frac{4}{x+3}\right)^2-3\\ & =\frac{16}{\left(x+3\right)^2}-3\\ & =\frac{16-3\left(x+3\right)^2}{\left(x+3\right)^2}\\ & =\frac{16-3\left(x^2+6x+9\right)}{\left(x+3\right)^2}\\ & =\frac{16-3x^2-18x-27}{\left(x+3\right)^2}\\ & =\frac{-3x^2-18x-11}{\left(x+3\right)^2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\ \end{align*}

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Category Archives: Calculus

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 1

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 4}\left(\frac{x^3-64}{x^2-16}\right)$

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 8

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 0}\left(\frac{3x+2}{x^2-2x+4}\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 7

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{\sqrt{3x}}{x\sqrt{x+1}}\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 6

PROBLEM:

Evaluate $\displaystyle \lim_{x\to 2}\left(4x-3\right)\left(x^2+5\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 5

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 8}\left(2x+\sqrt[3]{x}-4\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 4

PROBLEM:

Evaluate $\displaystyle \lim\limits _{x\to \frac{\pi }{3}}\left(\frac{\sin\:2x}{\sin\:x}\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 3

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 2

PROBLEM:

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.2, Problem 1

PROBLEM:

Evaluate $\displaystyle \lim _{x\to 2}\left(x^2-4x+3\right)$ .

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1, Problem 10

PROBLEM:

If $\displaystyle f\left(x\right)=\frac{4}{x+3}$ and $\displaystyle \:g\left(x\right)=x^2-3$ , find $\displaystyle f\left[g\left(x\right)\right]$ and $\displaystyle g\left[f\left(x\right)\right]$ .

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 4}\left(\frac{x^3-64}{x^2-16}\right)

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 0}\left(\frac{3x+2}{x^2-2x+4}\right).

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 3}\left(\frac{\sqrt{3x}}{x\sqrt{x+1}}\right).

PROBLEM:

Evaluate \displaystyle \lim_{x\to 2}\left(4x-3\right)\left(x^2+5\right).

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 8}\left(2x+\sqrt[3]{x}-4\right).

PROBLEM:

Evaluate \displaystyle \lim\limits _{x\to \frac{\pi }{3}}\left(\frac{\sin\:2x}{\sin\:x}\right).

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right).

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right).

PROBLEM:

Evaluate \displaystyle \lim _{x\to 2}\left(x^2-4x+3\right).

PROBLEM:

If \displaystyle f\left(x\right)=\frac{4}{x+3} and \displaystyle \:g\left(x\right)=x^2-3 , find \displaystyle f\left[g\left(x\right)\right] and \displaystyle g\left[f\left(x\right)\right].

Evaluate $\displaystyle \lim\limits_{x\to 4}\left(\frac{x^3-64}{x^2-16}\right)$

Evaluate $\displaystyle \lim\limits_{x\to 0}\left(\frac{3x+2}{x^2-2x+4}\right)$ .

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{\sqrt{3x}}{x\sqrt{x+1}}\right)$ .

Evaluate $\displaystyle \lim_{x\to 2}\left(4x-3\right)\left(x^2+5\right)$ .

Evaluate $\displaystyle \lim\limits_{x\to 8}\left(2x+\sqrt[3]{x}-4\right)$ .

Evaluate $\displaystyle \lim\limits _{x\to \frac{\pi }{3}}\left(\frac{\sin\:2x}{\sin\:x}\right)$ .

Evaluate $\displaystyle \lim\limits_{x\to \frac{\pi }{4}}\left(\tan\:x+\sin\:x\right)$ .

Evaluate $\displaystyle \lim\limits_{x\to 3}\left(\frac{4x+2}{x+4}\right)$ .

Evaluate $\displaystyle \lim _{x\to 2}\left(x^2-4x+3\right)$ .

If $\displaystyle f\left(x\right)=\frac{4}{x+3}$ and $\displaystyle \:g\left(x\right)=x^2-3$ , find $\displaystyle f\left[g\left(x\right)\right]$ and $\displaystyle g\left[f\left(x\right)\right]$ .