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.

Understand how to evaluate the limit of 4x^21 - e^x as x approaches 0 step-by-step. Learn the definition of limits and the behavior of exponential functions in limits. Suitable for undergraduate students and anyone interested in calculus and limits.

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