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 $\underset{x\to 2}{lim}\frac{\frac{1}{x+1}-\frac{1}{3}}{x-2}$ is in the form of the definition of the derivative $f^{\prime }(a)=\underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}$.
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 $f^{\prime }(x)$ is given by $\underset{x\to 2}{lim}\frac{f(x)-f(2)}{x-2}=\underset{x\to 2}{lim}\frac{\frac{1}{x+1}-\frac{1}{3}}{x-2}$.
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: $\underset{x\to 2}{lim}\frac{\frac{2-x}{3(x+1)}}{x-2}$. Simplify and evaluate the limit.