MHB Limit of e to the tangent of x

[thereidisanceman]

lim {e}^{tan(x)}
x\implies\{(pi/2)}^{+}

How do I start? Do I plug in and if so what next?

 

[MarkFL]

I would first consider what we get for:

$$\lim_{x\to\frac{\pi}{2}^{+}}\tan(x)$$

What is the value of the above limit?

 

[thereidisanceman]

tan (\pi/2)

 

[MarkFL]

tan (\pi/2)

Think of the graph of:

$$y=\tan(x)$$

Where is the curve going as we approach $$x=\frac{\pi}{2}$$ from the right side?

 

[thereidisanceman]

oh, -\infty from the right

 

[MarkFL]

oh, -\infty from the right

Yes, and so what is $e$ raised to this power?

Note: to get your $\LaTeX$ code to parse correctly, wrap it in the MATH tags. Click the $$\sum$$ button on our toolbar, and then place your code in between the generated tags. :D

 

[thereidisanceman]

0?

$$\sin\left({\lim_{{}\to{a}}}\right)$$
(not related)

 

[MarkFL]

Yes, because:

$$\lim_{x\to-\infty}e^x=0$$ :D

 

[thereidisanceman]

Oh coolio, thanks!

 

[Ackbach]

Also, if you right-click on any correctly displaying $\LaTeX$ math, choose Show Math As -> TeX Commands, you will see what someone typed into get a particular output. You can also use single-dollar signs to enclose displayed math, like this: $\int_{-\infty}^{\infty}e^{-x^2} \, dx=\sqrt{\pi}$, which I typed up using

$\int_{-\infty}^{\infty}e^{-x^2} \, dx=\sqrt{\pi}$,

or you can use double-dollar signs to get a displayed equation like this:

$$\int_{-\infty}^{\infty}e^{-x^2} \, dx=\sqrt{\pi},$$ which I typed up using

$$\int_{-\infty}^{\infty}e^{-x^2} \, dx=\sqrt{\pi}.$$

Notice the difference in size of the various symbols.

 

[Greg]

$$\lim_{x\to\frac{\pi}{2}^+}e^{\tan(x)}$$Over the interval $\left(\frac{\pi}{2},\pi\right]$ we have $e^{\tan(x)}=\frac{1}{e^{|\tan(x)|}}$, hence$$\lim_{x\to\frac{\pi}{2}^+}e^{\tan(x)}=\lim_{x\to\frac{\pi}{2}^+}\frac{1}{e^{|\tan(x)|}}=0$$

 

[Maged Saeed]

Also , you can see the graph of the function where (tanx = - infinity) when it approaches n(pi/2) or -n(pi\2), therefore the function will go to zero.
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