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