Talon Cannon

2023-03-25

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

paumEFupbom2lda

Beginner2023-03-26Added 6 answers

We have

$L=\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}$

To apply L'Hôpital's rule, we must have a $0\text{/}0$ or $\infty \text{/}\infty$ situation. If we plug in $t=0$ we find that:

$L=\frac{{e}^{0}-1}{0}=\frac{0}{0}$

So, we can apply the L'Hôpital's rule, which says:

$L=\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}=\frac{\frac{\text{d}}{\text{d}t}({e}^{t}-1)}{\frac{\text{d}}{\text{d}t}t}$

We know that $e}^{x$ is one of the functions with the property that $f\prime \left(x\right)=f\left(x\right)$, and as $-1$ is just a constant, it will vanish when we take the derivative.

$\therefore L=\underset{t\to 0}{lim}\frac{{{e}^{t}}}{{1}}=\underset{t\to 0}{lim}{{e}^{t}}$

$L={e}^{0}=1$

$L=\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}$

To apply L'Hôpital's rule, we must have a $0\text{/}0$ or $\infty \text{/}\infty$ situation. If we plug in $t=0$ we find that:

$L=\frac{{e}^{0}-1}{0}=\frac{0}{0}$

So, we can apply the L'Hôpital's rule, which says:

$L=\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}=\frac{\frac{\text{d}}{\text{d}t}({e}^{t}-1)}{\frac{\text{d}}{\text{d}t}t}$

We know that $e}^{x$ is one of the functions with the property that $f\prime \left(x\right)=f\left(x\right)$, and as $-1$ is just a constant, it will vanish when we take the derivative.

$\therefore L=\underset{t\to 0}{lim}\frac{{{e}^{t}}}{{1}}=\underset{t\to 0}{lim}{{e}^{t}}$

$L={e}^{0}=1$

Jase Leonard

Beginner2023-03-27Added 6 answers

$\underset{t\to 0}{lim}\frac{{e}^{t}-1}{t}{=}_{DLH}^{\left(\frac{0}{0}\right)}\underset{t\to 0}{lim}{e}^{t}={e}^{0}=1$

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 $e}^{-2x$?

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 $\frac{1+x}{1+{x}^{2}}dx$?

What is the limit as x approaches negative infinity of $x+\sqrt{{x}^{2}+2x}$?