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.