The graphs above illustrate the comparison between the actual functions and their corresponding Taylor polynomials for $n = 5$:

1. For $a^x$ with $a = 2$ (left plot), the Taylor series approximation closely follows the exponential curve, particularly near $x = 0$, but diverges for larger $x$.

2. For $\frac{1}{1+x}$ (right plot), the Taylor series accurately represents the function near $x = 0$, but starts to deviate significantly as $x$ approaches $-1$ (where the function has a vertical asymptote) and for large $x$.

These visualizations help illustrate how Taylor series approximate functions and where they start to lose accuracy as $x$ moves away from the expansion point.

Would you like a more detailed explanation or further questions about these concepts?

Here are five related questions to deepen understanding:

1. How does the convergence of a Taylor series depend on the degree $n$?
2. Why does the Taylor series for $\frac{1}{1+x}$ diverge near $x = -1$?
3. How would the Taylor series for $a^x$ behave for different values of $a$?
4. What happens to the accuracy of the Taylor polynomial if we increase $n$?
5. Can you explain why the Taylor series for $1/(1+x)$ is geometric?

Tip: Taylor series are best near the point of expansion (usually $x = 0$). Farther away, higher degrees are needed for accuracy.