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

To evaluate the limit: \[ \lim_{x \rightarrow +\infty} x \cdot 2^{x} \] **Analysis:** 1. **Exponential Growth vs. Polynomial Growth:** - The function \( 2^x \) is an exponential function, which grows **much faster** than any polynomial function of \( x \). In this case, \( x \) is a linear (first-degree polynomial) function. - As \( x \) becomes very large, the exponential term \( 2^x \) will dominate the linear term \( x \). 2. **Applying the Limit:** - Since \( 2^x \) grows exponentially without bound as \( x \rightarrow +\infty \), multiplying it by \( x \) (which also tends to infinity, but much more slowly) will still result in an expression that grows without bound. 3. **Conclusion:** - Therefore, the limit is: \[ \lim_{x \rightarrow +\infty} x \cdot 2^{x} = +\infty \] **Intuitive Understanding:** - Imagine \( x = 10 \): \( x \cdot 2^{x} = 10 \cdot 1024 = 10,240 \) - For \( x = 20 \): \( x \cdot 2^{x} = 20 \cdot 1,048,576 = 20,971,520 \) - As \( x \) increases, the value of \( 2^x \) increases so rapidly that even when multiplied by \( x \), the product still tends to infinity. **Final Answer:** \[ \lim_{x \rightarrow +\infty} x \cdot 2^{x} = +\infty \]