To evaluate the limit

$\lim_{{x \to 2}} \frac{(x - 3)^{10} - 1}{4 - x^2}$

we can proceed with the following steps:

Step 1: Check for Direct Substitution

By substituting $x = 2$ directly:

• The numerator becomes $(2 - 3)^{10} - 1 = (-1)^{10} - 1 = 1 - 1 = 0$.
• The denominator becomes $4 - 2^2 = 4 - 4 = 0$.

This results in a $\frac{0}{0}$ indeterminate form, so we need to simplify further.

Step 2: Factor the Denominator

Observe that $4 - x^2$ can be factored as: $4 - x^2 = (2 - x)(2 + x)$

Step 3: Apply the Binomial Expansion for the Numerator

Since $(x - 3)^{10} - 1$ is a difference of two terms, we can consider expanding it. However, this can be quite complex to expand directly. A more efficient approach might be using L'Hôpital's Rule because it is a $\frac{0}{0}$ indeterminate form.

Step 4: Apply L'Hôpital's Rule

Differentiate the numerator and the denominator with respect to $x$:

1. Differentiate the numerator $(x - 3)^{10} - 1$: $\frac{d}{dx} \left( (x - 3)^{10} - 1 \right) = 10(x - 3)^9$

2. Differentiate the denominator $4 - x^2$: $\frac{d}{dx} \left( 4 - x^2 \right) = -2x$

Now the limit becomes: $\lim_{{x \to 2}} \frac{10(x - 3)^9}{-2x}$

Step 5: Substitute $x = 2$ in the Simplified Expression

Substituting $x = 2$ into the simplified expression gives: $\frac{10(2 - 3)^9}{-2 \cdot 2} = \frac{10(-1)^9}{-4} = \frac{-10}{-4} = \frac{10}{4} = \frac{5}{2}$

Final Answer

$\lim_{{x \to 2}} \frac{(x - 3)^{10} - 1}{4 - x^2} = \frac{5}{2}$

Would you like further details or have any questions?

Here are five related questions you might find helpful:

1. How would you solve this limit if the exponent in the numerator was different?
2. Can L'Hôpital's Rule always be applied in cases of indeterminate forms?
3. How would the answer change if the exponent were an even number?
4. What are the conditions necessary to apply L'Hôpital's Rule?
5. How does the binomial expansion help in solving limit problems?

Tip: When facing an indeterminate form, always check if factoring, expanding, or L'Hôpital’s Rule can simplify the expression.