The limit you provided can be interpreted as:

$\lim_{x \to 0} 4x^{21} - e^x$

Let's evaluate this limit step by step.

Step 1: Evaluate $\lim_{x \to 0} 4x^{21}$

As $x$ approaches 0, $4x^{21}$ will approach 0 because $x^{21}$ becomes very small (tends to 0) and multiplying by 4 doesn't change this.

$\lim_{x \to 0} 4x^{21} = 0$

Step 2: Evaluate $\lim_{x \to 0} e^x$

As $x$ approaches 0, $e^x$ approaches $e^0 = 1$.

$\lim_{x \to 0} e^x = 1$

Step 3: Combine the results

Now, subtract the limits:

$\lim_{x \to 0} \left(4x^{21} - e^x\right) = 0 - 1 = -1$

Final Answer:

$\lim_{x \to 0} 4x^{21} - e^x = -1$

Would you like more details or have any questions?

Here are 5 related questions:

1. What happens to the limit if the power of $x$ is changed in $4x^{21}$?
2. How does the behavior of $e^x$ change as $x$ approaches a different value, like $\infty$?
3. What if we replace $e^x$ with a different exponential function, like $e^{-x}$?
4. Can you explain the limit $\lim_{x \to 0} \frac{e^x - 1}{x}$?
5. How would the limit change if there were additional terms involving higher powers of $x$?

Tip: When evaluating limits, it’s often helpful to break down the expression into simpler parts, evaluate each part individually, and then combine the results.