Tag Archives: Feliciano and Uy

Differential and Integral Calculus by Feliciano and Uy, Exercise 1.3, Problem 2

Advertisements

PROBLEM:

Evaluate \displaystyle \lim\limits_{x\to 2}\left(\frac{x^2+2x-8}{3x-6}\right)

Advertisements

SOLUTION:

A straight substitution of x=2 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 2}\left(\frac{x^2+2x-8}{3x-6}\right)& =\lim\limits_{x\to 2}\left(\frac{\left(x+4\right)\left(x-2\right)}{3\left(x-2\right)}\right)\\
\\
&=\lim\limits_{x\to 2}\left(\frac{x+4}{3}\right)\\
\\
&=\frac{2+4}{3}\\
\\
&=\frac{6}{3}\\
\\
& =2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\
\\
\end{align*}
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

Advertisements

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*}
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

Advertisements

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*}
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

Advertisements

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*}
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

Advertisements

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*}
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

Advertisements

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*}
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

Advertisements

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*}
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

Advertisements

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*}
Advertisements
Advertisements