Evaluate Lim_(nrarr oo)(tan[(pi-4)/(4)+(1+(1)/(n))^(a)])^(n)(alpha inQ)

Simplify Expression: First, let's simplify the expression inside the tangent function. We have: tan[(π-4)/(4) + (1 + 1/n)^(a)] Since α is a rational number, we can denote it as α = p/q where p and q are integers. The expression (1 + 1/n)^(a) can be approximated as 1 when n approaches infinity, because (1 + 1/n) tends to 1 and any power of 1 is still 1. So, the expression inside the tangent function simplifies to: tan[(π-4)/(4) + 1]Calculate Angle: Now, let's calculate the exact value of the angle inside the tangent function: (π-4)/4 + 1 = π/4 - 1 + 1 = π/4 So, the expression simplifies to: tan(π/4) We know that tan(π/4) = 1, so the expression inside the limit becomes: (1)^(n)Evaluate Limit: Since (1)^(n) is simply 1 for any value of n, the limit as n approaches infinity of (1)^(n) is just 1. Therefore, the limit of the original expression is: Lim_(n→∞)(tan[(π-4)/(4) + (1 + 1/n)^(a)])^(n) = 1

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Evaluate
Lim_(nrarr oo)(tan[(pi-4)/(4)+(1+(1)/(n))^(a)])^(n)(alpha inQ)

Evaluate $\operatorname{Lim}_{\mathrm{n} \rightarrow \infty}\left(\tan \left[\frac{\pi-4}{4}+\left(1+\frac{1}{\mathrm{n}}\right)^{a}\right]\right)^{\mathrm{n}}(\alpha \in \mathrm{Q})$

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate

\operatorname{Lim}_{\mathrm{n} \rightarrow \infty}\left(\tan \left[\frac{\pi-4}{4}+\left(1+\frac{1}{\mathrm{n}}\right)^{a}\right]\right)^{\mathrm{n}}(\alpha \in \mathrm{Q})

Simplify Expression: First, let's simplify the expression inside the tangent function. We have: $\newline$ $\tan\left[\frac{\pi-4}{4} + \left(1 + \frac{1}{n}\right)^{a}\right]$ $\newline$ Since $\alpha$ is a rational number, we can denote it as $\alpha = \frac{p}{q}$ where $p$ and $q$ are integers. The expression $\left(1 + \frac{1}{n}\right)^{a}$ can be approximated as $1$ when $n$ approaches infinity, because $\left(1 + \frac{1}{n}\right)$ tends to $1$ and any power of $1$ is still $1$ . So, the expression inside the tangent function simplifies to: $\newline$ $\alpha$ $2$
Calculate Angle: Now, let's calculate the exact value of the angle inside the tangent function: $\newline$ $(\pi-4)/4 + 1 = \pi/4 - 1 + 1 = \pi/4$ $\newline$ So, the expression simplifies to: $\newline$ $\tan(\pi/4)$ $\newline$ We know that $\tan(\pi/4) = 1$ , so the expression inside the limit becomes: $\newline$ $(1)^{(n)}$
Evaluate Limit: Since $(1)^{(n)}$ is simply $1$ for any value of $n$ , the limit as $n$ approaches infinity of $(1)^{(n)}$ is just $1$ . Therefore, the limit of the original expression is: $\lim_{n\to\infty}(\tan\left[\frac{\pi-4}{4} + (1 + \frac{1}{n})^{(a)}\right])^{(n)} = 1$