majmunomog

2022-09-03

$$\underset{p\to 0}{lim}\frac{1}{2p}((1+p){e}^{-\frac{y}{1+p}}-(1-p){e}^{-\frac{y}{1-p}})={e}^{-y}+y{e}^{-y}$$

I have already tried L'Hopital's Rule, but it gave me something that I couldn't simplify. The problem seems to be the $\frac{1}{2p}$ term never seems to go away. I know the exponential function can be represented as: ${e}^{x}=\underset{n\to \mathrm{\infty}}{lim}(1+\frac{x}{n}{)}^{n}$, but it doesn't seem immediately obvious how that would apply in this situation.

I have already tried L'Hopital's Rule, but it gave me something that I couldn't simplify. The problem seems to be the $\frac{1}{2p}$ term never seems to go away. I know the exponential function can be represented as: ${e}^{x}=\underset{n\to \mathrm{\infty}}{lim}(1+\frac{x}{n}{)}^{n}$, but it doesn't seem immediately obvious how that would apply in this situation.

Shania Delacruz

Beginner2022-09-04Added 7 answers

HINT

It has form $\text{}\underset{\mathrm{x}\to 0}{lim}\text{}\frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(-\mathrm{x})}{\mathrm{x}-(-\mathrm{x})}\phantom{\rule{mediummathspace}{0ex}}.\text{}$. Relate that to a derivative.

Note that this solution by recognizing the limit as a derivative employs only knowledge of the definition of the derivative and the basic rules for calculating derivatives of polynomials and powers. It does not require knowledge of more advanced techniques such as power series or Taylor series, l'Hôpital's rule, the mean-value theorem, etc.

It has form $\text{}\underset{\mathrm{x}\to 0}{lim}\text{}\frac{\mathrm{f}(\mathrm{x})-\mathrm{f}(-\mathrm{x})}{\mathrm{x}-(-\mathrm{x})}\phantom{\rule{mediummathspace}{0ex}}.\text{}$. Relate that to a derivative.

Note that this solution by recognizing the limit as a derivative employs only knowledge of the definition of the derivative and the basic rules for calculating derivatives of polynomials and powers. It does not require knowledge of more advanced techniques such as power series or Taylor series, l'Hôpital's rule, the mean-value theorem, etc.

bolton8l

Beginner2022-09-05Added 1 answers

Hint: Writing the numerator of the fraction in the limit as

$$p({e}^{-\frac{y}{1+p}}+{e}^{-\frac{y}{1-p}})+{e}^{-\frac{y}{1+p}}-{e}^{-\frac{y}{1-p}}$$

should help.

$$p({e}^{-\frac{y}{1+p}}+{e}^{-\frac{y}{1-p}})+{e}^{-\frac{y}{1+p}}-{e}^{-\frac{y}{1-p}}$$

should help.

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?