zakinutuzi

2021-12-17

Find a region whose size matches the specified limit. Do not evaluate the limit. lim x tends to infinity summation $\frac{\pi }{4n}×\mathrm{tan}\frac{i\pi }{4n}$

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Philip Williams

Expert

Step 1
Remember that

Where $\mathrm{\Delta }x=\frac{b-a}{n}$
Step 2
$\underset{n⇒\mathrm{\infty }}{lim}\sum _{i=1}^{n}\frac{\pi }{4n}\mathrm{tan}\left(\frac{i\pi }{4n}\right)$
Comparing shows that
$\mathrm{\Delta }x=\frac{\pi }{4n}$
$a=0$
$\mathrm{\Delta }x=\frac{b-a}{n}=\frac{\pi }{4n}⇒b=\frac{\pi }{4}$
and $f\left(x\right)=\mathrm{tan}x$
Therefore,

We determine that
This sum represent the area under the curve $y=\mathrm{tan}x$ and above x-axis in the interval
The sum represent the area of the blue region in the graph below:

Still Have Questions?

ol3i4c5s4hr

Expert

Solution:
Given: $\underset{n⇒\mathrm{\infty }}{lim}\sum _{i=1}^{n}\frac{\pi }{4n}\mathrm{tan}\frac{\pi }{4n}$

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