Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim x🠂2) 1/x+1 - 1/3 / x-2

Verified step by step guidance

1. Step 1: Recognize that the given limit represents the derivative of a function at a point. The expression \( \lim_{{x \to 2}} \frac{\frac{1}{x+1} - \frac{1}{3}}{x-2} \) is in the form of the definition of the derivative \( f'(a) = \lim_{{x \to a}} \frac{f(x) - f(a)}{x-a} \).
2. Step 2: Identify the function \( f(x) \) and the point \( a \). From the expression \( \frac{1}{x+1} \), we can deduce that \( f(x) = \frac{1}{x+1} \). The point \( a \) is given by the limit \( x \to 2 \), so \( a = 2 \).
3. Step 3: Calculate \( f(a) \). Substitute \( a = 2 \) into \( f(x) \) to find \( f(2) = \frac{1}{2+1} = \frac{1}{3} \). This matches the \( \frac{1}{3} \) in the limit expression, confirming our function and point.
4. Step 4: Set up the derivative calculation. The derivative \( f'(x) \) is given by \( \lim_{{x \to 2}} \frac{f(x) - f(2)}{x-2} = \lim_{{x \to 2}} \frac{\frac{1}{x+1} - \frac{1}{3}}{x-2} \).
5. Step 5: Simplify the expression to find the limit. Combine the fractions in the numerator: \( \frac{1}{x+1} - \frac{1}{3} = \frac{3 - (x+1)}{3(x+1)} = \frac{2-x}{3(x+1)} \). Substitute this back into the limit: \( \lim_{{x \to 2}} \frac{\frac{2-x}{3(x+1)}}{x-2} \). Simplify and evaluate the limit.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, the limit helps determine the slope of the curve at a specific point, which is essential for understanding the behavior of the function near that point.

Derivatives

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this problem, finding the derivative at the point (a, f(a)) is crucial for determining the slope of the curve.

Rational Functions

Rational functions are ratios of polynomials, and they often require careful analysis when evaluating limits, especially when approaching points where the function may be undefined. In this question, the limit involves a rational expression, and understanding how to simplify or manipulate such expressions is key to finding the desired limit.