1-38= Find the limit. Use l'Hospital's Rule where...

$1-38=$ Find the limit. Use l'Hospital's Rule where appropriate.
If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.
$$\lim _{x \rightarrow 1} \frac{x^{a}-a x+a-1}{(x-1)^{2}}$$

Key Concepts

Taylor Series Expansion

Taylor series expansion is a technique used to express a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This method is particularly useful for approximating functions near that point and can simplify the evaluation of limits, especially when dealing with powers or polynomial-like structures in the numerator and denominator.

Higher-Order Derivatives

In cases where the first differentiation does not resolve the indeterminate form (for example, when the 0/0 form persists), higher-order derivatives become necessary. These derivatives, often encapsulated in the Taylor series expansion, provide a deeper level of function approximation that facilitates the evaluation of limits involving higher multiplicities of the vanishing factor.

Indeterminate Forms

When evaluating limits by direct substitution leads to expressions like 0/0, the limit is said to be in an indeterminate form. Recognizing such forms is essential because it signals that further analysis, such as applying l'Hôpital's Rule or using algebraic manipulation, is required to compute the limit accurately.

l'Hôpital's Rule

l'Hôpital's Rule provides a method to evaluate limits of indeterminate forms by differentiating the numerator and the denominator separately. The rule applies when both the numerator and denominator approach zero or both approach infinity as the limit is taken, and the resulting limit after differentiation is easier to compute.

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