The value of
\(\lim_{{x \to 1}} \frac{{(x^2 - 1) \sin^2(\pi x)}}{{x^4 - 2x^3 + 2x - 1}}\)
is equal to
The value of
\(\lim_{{x \to 1}} \frac{{(x^2 - 1) \sin^2(\pi x)}}{{x^4 - 2x^3 + 2x - 1}}\)
is equal to
\(\frac{π²}{6}\)
\(\frac{π²}{3}\)
\(\frac{π²}{2}\)
- π²
The Correct Option is D
Solution and Explanation
The correct answer is (D) : π²
\(\lim_{{x \to 1}} \frac{{(x^2 - 1) \sin^2(\pi x)}}{{x^4 - 2x^3 + 2x - 1}}\)
\(=\lim_{{x \to 1}} \frac{{(x+1)(x-1) \sin^2(\pi x)}}{{(x-1)^3(x+1)}}\)
Let x-1 = t
\(\lim_{{t \to 0}} \frac{{(2+t)t \sin^2(\pi t)}}{{t^3(t+2)}}\) \(= \) \(\lim_{{t \to 0}} \frac{{\sin^2(\pi t)}}{{\pi^2t^2}} \cdot \pi^2\) = \(\pi^2\)
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).
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