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Part 1
Evaluate the following limit. Use​ l'Hôpital's Rule when it is convenient and applicable.
ModifyingBelow lim With x right arrow minus 1StartFraction 7 x Superscript 4 Baseline plus 7 x cubed plus 4 x plus 4 Over x plus 1 EndFraction
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Part 1
Use​ l'Hôpital's Rule to rewrite the given limit so that it is not an indeterminate form.
ModifyingBelow lim With x right arrow minus 1StartFraction 7 x Superscript 4 Baseline plus 7 x cubed plus 4 x plus 4 Over x plus 1 EndFraction
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Part 2
Evaluate the limit.
ModifyingBelow lim With x right arrow minus 1StartFraction 7 x Superscript 4 Baseline plus 7 x cubed plus 4 x plus 4 Over x plus 1 EndFraction
equals

enter your response here ​(Type an exact​ answer.)

To evaluate the limit \( \lim_{{x \to -1}} \frac{{7x^4 + 7x^3 + 4x + 4}}{{x + 1}} \), we can use l'Hôpital's Rule if the limit is in an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

First, let's check the form of the limit by substituting \( x = -1 \):

\[ 
\text{Numerator: } 7(-1)^4 + 7(-1)^3 + 4(-1) + 4 = 7 - 7 - 4 + 4 = 0 
\]

\[ 
\text{Denominator: } (-1) + 1 = 0 
\]

Since both the numerator and the denominator are 0 when \( x = -1 \), the limit is in the indeterminate form \(\frac{0}{0}\). Thus, we can apply l'Hôpital's Rule, which states that:

\[ 
\lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} 
\]

if the original limit is in an indeterminate form. We need to differentiate the numerator and the denominator.

1. Differentiate the numerator: \( 7x^4 + 7x^3 + 4x + 4 \):

\[ 
\frac{d}{dx} (7x^4 + 7x^3 + 4x + 4) = 28x^3 + 21x^2 + 4 
\]

2. Differentiate the denominator: \( x + 1 \):

\[ 
\frac{d}{dx} (x + 1) = 1 
\]

Now, we apply l'Hôpital's Rule:

\[ 
\lim_{{x \to -1}} \frac{7x^4 + 7x^3 + 4x + 4}{x + 1} = \lim_{{x \to -1}} \frac{28x^3 + 21x^2 + 4}{1} 
\]

This simplifies to:

\[ 
\lim_{{x \to -1}} (28x^3 + 21x^2 + 4) 
\]

Now we substitute \( x = -1 \):

\[ 
28(-1)^3 + 21(-1)^2 + 4 = 28(-1) + 21(1) + 4 = -28 + 21 + 4 = -28 + 25 = -3 
\]

So, the limit is:

\[ 
\boxed{-3} 
\]

Would you like details or have any questions?

Here are some related questions you might find interesting:

1. What is l'Hôpital's Rule, and when is it applicable?
2. How do you determine if a limit is in an indeterminate form?
3. Can l'Hôpital's Rule be applied more than once? If so, when?
4. How do you differentiate polynomial functions?
5. What are some other methods to evaluate limits besides l'Hôpital's Rule?
6. Can l'Hôpital's Rule be applied to limits involving infinity?
7. How do you handle higher-order indeterminate forms?
8. What is the significance of the limit in calculus?

**Tip:** Always verify if a limit is in an indeterminate form before applying l'Hôpital's Rule, and ensure to simplify the expressions as much as possible for accurate results.

Learn how to evaluate the limit \( \lim_{x \to -1} \frac{7x^4 + 7x^3 + 4x + 4}{x + 1} \) using l'Hôpital's Rule. Step-by-step solution provided with differentiation of numerator and denominator. Suitable for college-level calculus students.

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