lim_(x rarr(pi)/(4))(sin(2x)-1)/(sin^(2)(2x)-1)

Simplify Expression: We are asked to find the limit of the function (sin(2x) - 1) / (sin^2(2x) - 1) as x approaches π/4. Let's first simplify the expression if possible before substituting the value of x.Factorize Difference of Squares: Notice that sin^2(2x) - 1 is a difference of squares which can be factored as (sin(2x) - 1)(sin(2x) + 1).Cancel Common Terms: Now we can rewrite the original expression by canceling out the common terms in the numerator and the denominator: (sin(2x) - 1) / (sin^2(2x) - 1) = (sin(2x) - 1) / ((sin(2x) - 1)(sin(2x) + 1)) = 1 / (sin(2x) + 1)Substitute x Value: Next, we substitute x = π/4 into the simplified expression: lim_(x -> π/4) 1 / (sin(2x) + 1) = 1 / (sin(2 * π/4) + 1) = 1 / (sin(π/2) + 1)Evaluate Limit: Since sin(π/2) is equal to 1, we have: 1 / (sin(π/2) + 1) = 1 / (1 + 1) = 1 / 2Final Result: Therefore, the limit of the function as x approaches π/4 is 1/2.

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lim_(x rarr(pi)/(4))(sin(2x)-1)/(sin^(2)(2x)-1)

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin (2 x)-1}{\sin ^{2}(2 x)-1}$

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Calculus

Evaluate definite integrals using the chain rule

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\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin (2 x)-1}{\sin ^{2}(2 x)-1}

Simplify Expression: We are asked to find the limit of the function $(\sin(2x) - 1) / (\sin^2(2x) - 1)$ as $x$ approaches $\pi/4$ . Let's first simplify the expression if possible before substituting the value of $x$ .
Factorize Difference of Squares: Notice that $\sin^2(2x) - 1$ is a difference of squares which can be factored as $(\sin(2x) - 1)(\sin(2x) + 1)$ .
Cancel Common Terms: Now we can rewrite the original expression by canceling out the common terms in the numerator and the denominator: $\newline$ $(\sin(2x) - 1) / (\sin^{2}(2x) - 1) = (\sin(2x) - 1) / ((\sin(2x) - 1)(\sin(2x) + 1)) = 1 / (\sin(2x) + 1)$
Substitute x Value: Next, we substitute $x = \frac{\pi}{4}$ into the simplified expression: $\newline$ $\lim_{x \to \frac{\pi}{4}} \frac{1}{\sin(2x) + 1} = \frac{1}{\sin(2 \cdot \frac{\pi}{4}) + 1} = \frac{1}{\sin(\frac{\pi}{2}) + 1}$
Evaluate Limit: Since $\sin(\frac{\pi}{2})$ is equal to $1$ , we have: $\frac{1}{(\sin(\frac{\pi}{2}) + 1)} = \frac{1}{(1 + 1)} = \frac{1}{2}$
Final Result: Therefore, the limit of the function as $x$ approaches $\pi/4$ is $1/2$ .