Questions: Evaluate the following limits symbolically using unit a) lim as x approaches infinity of (sin(1/x) - 4e^(-2x))

Solution Steps

To evaluate the limit as \( x \) approaches infinity for the given expression, we need to analyze the behavior of each term separately. As \( x \) approaches infinity, \( \frac{1}{x} \) approaches 0, and \( \sin \left(\frac{1}{x}\right) \) approaches \( \sin(0) \), which is 0. Similarly, \( e^{-2x} \) approaches 0 because the exponent is negative and grows larger. Therefore, the limit of the entire expression should be the difference of these two limits, which is 0.

As \(x\) approaches infinity, \(\frac{1}{x}\) approaches 0. Therefore, \(\sin\left(\frac{1}{x}\right)\) approaches \(\sin(0)\), which is 0.

As \(x\) approaches infinity, the exponent \(-2x\) becomes very large and negative, making \(e^{-2x}\) approach 0. Therefore, \(4e^{-2x}\) also approaches 0.

The limit of the expression \(\sin\left(\frac{1}{x}\right) - 4e^{-2x}\) as \(x \to \infty\) is the difference of the limits of the individual terms: \[ \lim_{x \to \infty} \left(\sin\left(\frac{1}{x}\right) - 4e^{-2x}\right) = \lim_{x \to \infty} \sin\left(\frac{1}{x}\right) - \lim_{x \to \infty} 4e^{-2x} = 0 - 0 = 0 \]

Final Answer