Use the definition of the derivative to evaluate the following limits.
${\lim }_{x\to 2}{\frac{5^x-25}{x-2}}_{}$

Verified step by step guidance

1. Recognize that the limit \( \lim_{x\to2}\frac{5^{x}-25}{x-2} \) is in the indeterminate form \( \frac{0}{0} \), which suggests using L'Hôpital's Rule or algebraic manipulation.
2. Rewrite the expression \( 5^x - 25 \) as \( 5^x - 5^2 \) to identify a common factor.
3. Factor the numerator as a difference of powers: \( 5^x - 5^2 = (5-5)(5^{x-1} + 5^{x-2} \cdot 5 + \ldots + 5^2) \).
4. Apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately: differentiate \( 5^x \) to get \( 5^x \ln(5) \) and \( x-2 \) to get 1.
5. Evaluate the new limit \( \lim_{x\to2}\frac{5^x \ln(5)}{1} \) by substituting \( x = 2 \) into the expression.

Key Concepts

Definition of the Derivative

The derivative of a function at a point is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is expressed as f'(a) = lim (h→0) [f(a+h) - f(a)] / h. This concept is fundamental for evaluating limits that represent the slope of the tangent line to the curve at a specific point.

Limit Evaluation

Limits are used to analyze the behavior of functions as they approach a certain point. In this context, we are interested in the limit as x approaches 2 for the expression (5^x - 25) / (x - 2). Evaluating this limit often involves techniques such as direct substitution, factoring, or applying L'Hôpital's Rule when the limit results in an indeterminate form.

Exponential Functions

Exponential functions, such as 5^x, are functions where a constant base is raised to a variable exponent. They exhibit unique properties, including rapid growth and specific limits as x approaches certain values. Understanding the behavior of exponential functions is crucial for evaluating limits involving expressions like 5^x, especially when determining continuity and differentiability at specific points.