Q. 1 - Calculus

Question

1. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. 

(a) True or False: If a limit has an indeterminate form, then that limit does not have a real number as its solution. 

(b) True or False: L’Hopital’s rule can be used to find the ˆ limit of any quotient $\frac{f\left(x\right)}{g\left(x\right)}\text{ as }x\to c$. 

(c) True or False: When using L’Hopital’s rule, you need ˆ to apply the quotient rule in the differentiation step. 

(d) True or False: L’Hopital’s rule applies only to limits as $x\to 0$ or as $x\to \infty$.

(e) True or False: L’Hopital’s rule applies only to limits of ˆ the indeterminate form $\frac{0}{0} \mathrm{or} \frac{\infty }{\infty }$.

(f) True or False: If $\underset{x\to 2}{lim} \ln \left(f\right(x\left)\right)=4$, then $\underset{x\to 2}{lim} \ln f\left(x\right)=\ln 4$.

(g) True or False: If $\underset{x\to 2}{lim} \ln \left(f\right(x\left)\right)=\infty$, then $\underset{x\to 2}{lim} f\left(x\right)=\infty$.

(h) True or False: If $\underset{x\to 2}{lim} \ln \left(f\right(x\left)\right)=-\infty$, then $\underset{x\to 2}{lim} f\left(x\right)=-\infty$. 

Step-by-Step Solution

Answer

Part (a). False

Part (b). False

Part (c). False

Part (d). False

Part (e). True

Part (f). False

Part (g). True

Part (h). False

Part (a) Step 1. Given information.

We have been given a statements

We have to determine whether each of the statements that follow is true or false and justify

Part (a)

This is a false statement.

Example:

$$\begin{gathered} \underset{x\to \mathrm{\infty }}{lim} \frac{\ln x}{x^2} \\ =\underset{x\to \mathrm{\infty }}{lim} \frac{\frac{1}{x}}{2x} \\ =\underset{x\to \mathrm{\infty }}{lim} \frac{1}{2x^2} \\ =0 \end{gathered}$$

The example shows it is not necessary for the limit to be real number as its solution if a limit has an intermediate.

Part (b)

This is a false statement.

Example: $\underset{x\to 1}{lim} \frac{x}{x-1}$

This example shows that L’Hopital’s rule can not be used in any quotient of form $\frac{f\left(x\right)}{g\left(x\right)}$ as $x\to c$ .

Part (c)

This is a false statement.

Example:

$$\begin{gathered} \underset{x\to \mathrm{\infty }}{lim} \frac{\ln x}{x} \\ =\underset{x\to \mathrm{\infty }}{lim} \frac{\frac{1}{x}}{1} \\ =\underset{x\to \mathrm{\infty }}{lim} \frac{1}{x} \\ =0 \end{gathered}$$

This example shows that it is not needed to apply the quotient rule in the differentiation step while using L'Hopital rule

Part (d)

This is a false statement.

Example:

$$\begin{gathered} \underset{x\to 1}{lim} \frac{x-1}{x^2-1} \\ =\underset{x\to 1}{lim} \frac{1}{2x} \\ =\frac{1}{2} \end{gathered}$$

This example shows that L'hopital does not apply only to limit as $x\to 0$ or as  $x\to \infty$    

Part (e)

This is a true statement.

L’Hopital’s rule is applied to the limits of the form $\frac{0}{0}$ and $\frac{\infty }{\infty }$. These forms are called indeterminate forms.

Part (f)

This is a false statement.

Example:

Let $f\left(x\right)=e^x$

$$\begin{gathered} \underset{x\to 2}{lim} \ln \left(f\right(x\left)\right) \\ =\underset{x\to 2}{lim} \ln e^{x^2} \\ =\underset{x\to 2}{lim} x^2\ln e \\ =4 \end{gathered}$$

And Also 

$$\begin{gathered} \underset{x\to 2}{lim} f\left(x\right) \\ =\underset{x\to 2}{lim} e^{x^2} \\ =e^{2^2} \\ =e^4 \end{gathered}$$

Thus the above examples that if$\underset{x\to 2}{\lim }\left(f\left(x\right)\right)=4$ the is is not necessary that $\underset{x\to 2}{\mathrm{limln}}\left(f\left(x\right)\right)=4$

Part (g)

This is a true statement. Because value of log is$\infty$only for $\infty$

thus $\underset{x\to 2}{\lim }\ln \left(f\left(x\right)\right)=\infty$ only for $f\left(x\right)=\infty$$f\left(x\right)=\infty$

And $\underset{x\to 2}{lim} (\infty )=\mathrm{\infty }$ 

Part (h)

This is a false statement.

Example:

let$f\left(x\right)=0$

$\underset{x\to 2}{lim} \ln \left(0\right)=-\infty$

but

$\underset{x\to 2}{\lim }0=0$

The above state mentment shows that 

if $\underset{x\to 2}{\lim }\ln \left(f\left(x\right)\right)=-\infty$then it not necessary that$\underset{x\to 2}{\lim }\left(f\left(x\right)\right)=-\infty$