Evaluate the following limits. If the limit does not exist, explain why. \( \begin{array}{lll}\text { (a) } \lim _{t \rightarrow-2}\left(\frac{2-|t|}{2+t}\right) & \text { (b) } \lim _{x \rightarrow 2^{-}} \frac{6-3 x}{|x-2|} & \text { (c) } \lim _{x \rightarrow 3} \frac{1}{|3-x|} \\ \text { (c) } \lim _{x \rightarrow 0}\left(x^{2} \cos \left(\frac{2}{x}+4\right)\right) & \text { (d) } \lim _{x \rightarrow 0} x^{2}\left(1+\sin \frac{\pi}{x}\right) & \text { (e) } \lim _{x \rightarrow 2^{+}} \sqrt{x-2} \cos \left(\frac{1}{x-2}\right) \\ \text { (f) } \lim _{x \rightarrow \infty} \sin \left(2 \tan ^{-1}(x)\right) & \text { (g) } \lim _{x \rightarrow \infty} \arctan e^{x} & \text { (h) } \lim _{x \rightarrow \infty} \arctan e^{-x}\end{array} \)

Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow +\infty}\left(\arctan\left(e^{-x}\right)\right)\) - step1: Rewrite the expression: \(\arctan\left(\lim _{x\rightarrow +\infty}\left(e^{-x}\right)\right)\) - step2: Calculate: \(\arctan\left(0\right)\) - step3: Calculate the trigonometric value: \(0\) Calculate the limit \( \lim_{x \rightarrow 2^{+}} \sqrt{x - 2} \cos \left(\frac{1}{x - 2}\right) \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow 2^{+}}\left(\sqrt{x-2}\times \cos\left(\frac{1}{x-2}\right)\right)\) - step1: Calculate: \(0\) Calculate the limit \( \lim_{x \rightarrow 0}\left(x^{2} \cos \left(\frac{2}{x} + 4\right)\right) \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow 0}\left(x^{2}\cos\left(\frac{2}{x}+4\right)\right)\) - step1: Calculate: \(0\) Calculate the limit \( \lim_{t \rightarrow -2}\left(\frac{2 - |t|}{2 + t}\right) \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{t\rightarrow -2}\left(\frac{2-\left|t\right|}{2+t}\right)\) - step1: Evaluate the left-hand and the right-hand limits: \(\begin{align}&\lim _{t\rightarrow -2^{-}}\left(\frac{2-\left|t\right|}{2+t}\right)\\&\lim _{t\rightarrow -2^{+}}\left(\frac{2-\left|t\right|}{2+t}\right)\end{align}\) - step2: Evaluate the left-hand limit: \(\begin{align}&1\\&1\end{align}\) - step3: Calculate: \(1\) Calculate the limit \( \lim_{x \rightarrow 2^{-}} \frac{6 - 3x}{|x - 2|} \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow 2^{-}}\left(\frac{6-3x}{\left|x-2\right|}\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow 2^{-}}\left(\frac{6-3x}{-x+2}\right)\) - step2: Reduce the fraction: \(\lim _{x\rightarrow 2^{-}}\left(3\right)\) - step3: Calculate: \(3\) Calculate the limit \( \lim_{x \rightarrow \infty} \sin \left(2 \tan^{-1}(x)\right) \). Evaluate the limit by following steps: - step0: Evaluate using transformations: \(\lim _{x\rightarrow +\infty}\left(\sin\left(2\arctan\left(x\right)\right)\right)\) - step1: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(2\sin\left(\arctan\left(x\right)\right)\cos\left(\arctan\left(x\right)\right)\right)\) - step2: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(2\sin\left(\arctan\left(x\right)\right)\times \frac{1}{\sqrt{1+x^{2}}}\right)\) - step3: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(2\times \frac{x}{\sqrt{1+x^{2}}}\times \frac{1}{\sqrt{1+x^{2}}}\right)\) - step4: Multiply the terms: \(\lim _{x\rightarrow +\infty}\left(\frac{2x}{1+x^{2}}\right)\) - step5: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(\frac{x}{x^{2}}\times \frac{2}{\frac{1}{x^{2}}+1}\right)\) - step6: Calculate: \(\lim _{x\rightarrow +\infty}\left(\frac{1}{x}\times \frac{2}{\frac{1}{x^{2}}+1}\right)\) - step7: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(\frac{1}{x}\right)\times \lim _{x\rightarrow +\infty}\left(\frac{2}{\frac{1}{x^{2}}+1}\right)\) - step8: Calculate: \(\lim _{x\rightarrow +\infty}\left(\frac{1}{x}\right)\times \frac{\lim _{x\rightarrow +\infty}\left(2\right)}{\lim _{x\rightarrow +\infty}\left(\frac{1}{x^{2}}+1\right)}\) - step9: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(\frac{1}{x}\right)\times \frac{2}{\lim _{x\rightarrow +\infty}\left(\frac{1}{x^{2}}+1\right)}\) - step10: Rewrite the expression: \(\lim _{x\rightarrow +\infty}\left(\frac{1}{x}\right)\times \frac{2}{1}\) - step11: Rewrite the expression: \(0\times \frac{2}{1}\) - step12: Calculate: \(0\times 2\) - step13: Calculate: \(0\) Calculate the limit \( \lim_{x \rightarrow 0} x^{2}\left(1 + \sin \frac{\pi}{x}\right) \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow 0}\left(x^{2}\left(1+\sin\left(\frac{\pi }{x}\right)\right)\right)\) - step1: Calculate: \(0\) Calculate the limit \( \lim_{x \rightarrow 3} \frac{1}{|3 - x|} \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow 3}\left(\frac{1}{\left|3-x\right|}\right)\) - step1: Evaluate the left-hand and the right-hand limits: \(\begin{align}&\lim _{x\rightarrow 3^{-}}\left(\frac{1}{\left|3-x\right|}\right)\\&\lim _{x\rightarrow 3^{+}}\left(\frac{1}{\left|3-x\right|}\right)\end{align}\) - step2: Evaluate the left-hand limit: \(\begin{align}&+\infty\\&+\infty\end{align}\) - step3: Calculate: \(+\infty\) Calculate the limit \( \lim_{x \rightarrow \infty} \arctan e^{x} \). Evaluate the limit by following steps: - step0: Evaluate the limit: \(\lim _{x\rightarrow +\infty}\left(\arctan\left(e^{x}\right)\right)\) - step1: Rewrite the expression: \(\arctan\left(\lim _{x\rightarrow +\infty}\left(e^{x}\right)\right)\) - step2: Calculate: \(\arctan\left(+\infty\right)\) - step3: Calculate the trigonometric value: \(\frac{\pi }{2}\) Let's evaluate each limit step by step and explain the results: ### (a) \( \lim _{t \rightarrow -2}\left(\frac{2-|t|}{2+t}\right) \) 1. Substitute \( t = -2 \): \[ \frac{2 - |-2|}{2 + (-2)} = \frac{2 - 2}{0} = \frac{0}{0} \] This is an indeterminate form. 2. Simplifying: \[ |t| = -t \text{ for } t < 0 \Rightarrow \frac{2 - (-t)}{2 + t} = \frac{2 + t}{2 + t} = 1 \text{ for } t \neq -2 \] 3. Therefore, the limit is: \[ \lim_{t \rightarrow -2} 1 = 1 \] ### (b) \( \lim _{x \rightarrow 2^{-}} \frac{6-3x}{|x-2|} \) 1. As \( x \) approaches \( 2 \) from the left, \( |x - 2| = 2 - x \): \[ \lim_{x \rightarrow 2^{-}} \frac{6 - 3x}{2 - x} \] 2. Substitute \( x = 2 \): \[ \frac{6 - 3(2)}{2 - 2} = \frac{0}{0} \] This is an indeterminate form. 3. Simplifying: \[ \frac{6 - 3x}{2 - x} = \frac{-3(x - 2)}{-(x - 2)} = 3 \text{ for } x \neq 2 \] 4. Therefore, the limit is: \[ \lim_{x \rightarrow 2^{-}} 3 = 3 \] ### (c) \( \lim _{x \rightarrow 3} \frac{1}{|3-x|} \) 1. As \( x \) approaches \( 3 \): \[ |3 - x| = 3 - x \text{ for } x < 3 \text{ and } x - 3 \text{ for } x > 3 \] 2. The limit approaches: \[ \lim_{x \rightarrow 3} \frac{1}{|3 - x|} = \lim_{x \rightarrow 3} \frac{1}{0} = +\infty \] ### (d) \( \lim _{x \rightarrow 0}\left(x^{2} \cos \left(\frac{2}{x}+4\right)\right) \) 1. As \( x \) approaches \( 0 \): \[ x^{2} \text{ approaches } 0 \text{ and } \cos \left(\frac{2}{x}+4\right) \text{ oscillates between } -1 \text{ and } 1. \] 2. Therefore, the limit is: \[ 0 \cdot \text{(bounded oscillation)} = 0 \] ### (e) \( \lim _{x \rightarrow 0} x^{2}\left(1+\sin \frac{\pi}{x}\right) \) 1. As \( x \) approaches \( 0 \): \[ x^{2} \text{ approaches } 0 \text{ and } \sin \frac{\pi}{x} \text{ oscillates between } -1 \text{ and } 1. \] 2. Therefore, the limit is: \[ 0 \cdot \text{(bounded oscillation)} = 0 \] ### (f) \( \lim _{x \rightarrow 2^{+}} \sqrt{x-2} \cos \left(\frac{1}{x-2}\right) \) 1. As \( x \) approaches \( 2 \) from the right: \[ \sqrt{x - 2} \text{ approaches } 0 \text{ and } \cos \left(\frac{1}{x - 2}\right) \text{ oscillates between } -1 \text{ and } 1. \] 2. Therefore, the limit is: \[ 0 \cdot \text{(bounded oscillation)} = 0 \] ### (g) \( \lim _{x \rightarrow \infty} \sin \left(2 \tan^{-1}(x)\right) \) 1. As \( x \) approaches \( \infty \): \[ \tan^{-1}(x) \text{ approaches } \frac{\pi}{2} \Rightarrow 2 \tan^{-1}(x) \text{ approaches } \pi. \] 2. Therefore, the limit is: \[ \sin(\pi) = 0 \] ### (h) \( \lim _{x \rightarrow \infty} \arctan e^{x} \) 1. As \( x \) approaches \( \infty \): \[ e^{x} \text{ approaches } \infty \Rightarrow \arctan(e^{x}) \text{ approaches } \frac{\pi}{2}. \] ### (i) \( \lim _{x \rightarrow \infty} \arctan e^{-x} \) 1. As \( x \) approaches \( \infty \): \[ e^{-x} \text{ approaches } 0 \Rightarrow \arctan(e^{-x}) \text{ approaches } 0. \] ### Summary of Results: - (a) \( 1 \) - (b) \( 3 \) - (c) \( +\infty \) - (d) \( 0 \) - (e) \( 0 \) - (f) \( 0 \) - (g) \( 0 \) - (h) \( \frac{\pi}{2} \) - (i) \( 0 \)