Limit of e to the tangent of x

In summary, the limit of $e$ raised to the tangent of $x$ as $x$ approaches $\frac{\pi}{2}$ from the right is equal to 0, because the tangent function goes to negative infinity in that interval, causing the entire function to approach 0. This can also be seen from the graph of the function.
  • #1
thereidisanceman
8
0
lim {e}^{tan(x)}
x\implies\{(pi/2)}^{+}

How do I start? Do I plug in and if so what next?
 
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  • #2
I would first consider what we get for:

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

What is the value of the above limit?
 
  • #3
tan (\pi/2)
 
  • #4
thereidisanceman said:
tan (\pi/2)

Think of the graph of:

\(\displaystyle y=\tan(x)\)

Where is the curve going as we approach \(\displaystyle x=\frac{\pi}{2}\) from the right side?
 
  • #5
oh, -\infty from the right
 
  • #6
thereidisanceman said:
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 \(\displaystyle \sum\) button on our toolbar, and then place your code in between the generated tags. :D
 
  • #7
0?

\(\displaystyle \sin\left({\lim_{{}\to{a}}}\right)\)
(not related)
 
  • #8
Yes, because:

\(\displaystyle \lim_{x\to-\infty}e^x=0\) :D
 
  • #9
Oh coolio, thanks!
 
  • #10
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

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

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

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

Notice the difference in size of the various symbols.
 
  • #11
$$\lim_{x\to\frac{\pi}{2}^+}e^{\tan(x)}$$Over the interval $\left(\frac{\pi}{2},\pi\right]$ we have $e^{\tan(x)}=\frac{1}{e^{|\tan(x)|}}$, hence$$\lim_{x\to\frac{\pi}{2}^+}e^{\tan(x)}=\lim_{x\to\frac{\pi}{2}^+}\frac{1}{e^{|\tan(x)|}}=0$$
 
  • #12
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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FAQ: Limit of e to the tangent of x

What is the limit of e to the tangent of x as x approaches infinity?

The limit of e to the tangent of x as x approaches infinity is undefined. This is because the tangent function oscillates between positive and negative infinity as x approaches infinity, and the exponential function e^x grows without bound as x increases.

How do you find the limit of e to the tangent of x as x approaches a specific value?

To find the limit of e to the tangent of x as x approaches a specific value, you can use the L'Hopital's rule, which states that the limit of a quotient of two differentiable functions is equal to the limit of their derivatives if the original limit is of the form 0/0 or ∞/∞.

Can the limit of e to the tangent of x be negative?

Yes, the limit of e to the tangent of x can be negative. This can happen when the angle x is in the second or fourth quadrant, where the tangent function is negative. The exponential function e^x can also be negative for certain values of x, resulting in a negative limit.

Is there a way to simplify the limit of e to the tangent of x?

No, there is no simple way to simplify the limit of e to the tangent of x. However, you can use the product rule for limits to rewrite the limit as the product of two simpler limits, one for e^x and one for tangent x. This can sometimes make it easier to evaluate the limit.

What is the relationship between the limit of e to the tangent of x and the limit of tangent x as x approaches 0?

The limit of e to the tangent of x and the limit of tangent x as x approaches 0 are equal. This is because the tangent function can be rewritten as the quotient of sine and cosine, and the limit of sine x over cosine x as x approaches 0 is equivalent to the limit of tangent x as x approaches 0.

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