(b) \( \lim _{x \rightarrow+\infty} x \cdot 2^{x} \)

As \( x \) approaches positive infinity, the term \( 2^x \) grows exponentially, much faster than the linear growth of \( x \). Therefore, when you multiply \( x \) by \( 2^x \), the exponential part dominates, and the entire expression \( x \cdot 2^x \) heads towards infinity. In simpler terms, think of \( x \) as a little bystander while \( 2^x \) races away to infinity—eventually, \( x \) can't keep up! For mathematical rigor, you can apply L'Hôpital's Rule, which is often a handy technique for evaluating limits involving the form \( \infty \cdot 0 \). By rewriting the limit as \( \lim_{x \to +\infty} \frac{x}{\frac{1}{2^x}} \) and differentiating the top and bottom, it becomes clear that the exponential function in the denominator will shrink dramatically, confirming that the limit is indeed \( +\infty \).