Find the limit lim_{brightarrow1^-}int_0^bfrac{dx}{sqrt{1-x^2}}

Find the limit lim_{brightarrow1^-}int_0^bfrac{dx}{sqrt{1-x^2}}

Question
Limits and continuity
asked 2021-02-01
Find the limit
\(\lim_{b\rightarrow1^-}\int_0^b\frac{dx}{\sqrt{1-x^2}}\)

Answers (1)

2021-02-02
Given:
\(\lim_{b\rightarrow1^-}\int_0^b\frac{dx}{\sqrt{1-x^2}}\)
First, we will find integration:
\(\lim_{b\rightarrow1^-}\int_0^b\frac{dx}{\sqrt{1-x^2}}\)
This is an standart integral:
\(=[\arcsin(x)]_0^b\)
\(=[\arcsin(b)-\arcsin(0)]\)
\(=\arcsin(b)\)
Now, find the limit:
\(\lim_{b\rightarrow1^-}\arcsin(b)\)
Plug b=1
\(=\arcsin(1)\)
\(=\frac{\pi}{2}\)
\(\lim_{b\rightarrow1^-}\int_0^b\frac{dx}{\sqrt{1-x^2}}=\frac{\pi}{2}\)
0

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