The following limits represent the slope of a curve y = f(x) at the point (a, f(x)). Determine a function f and a number a; then, calculate the limit.

\(\lim _{x \rightarrow 2}\ \frac{\frac{1}{x+1}-\frac{1}{3}}{x-2}\)

SOLUTION STEP 1 Given is the slope of the curve y=f(x) Thus it is the derivative of the function f(x). 1 1 ie,f(x) = lim x+13 x2 x2 Thus comparing with the definition of the derivative f(x)f(a) f(x) = lim xa xa Thus we can find that f(x) = 1 and a = 2 x+1 Thus the given function is the slope of a curve y=f(x) at point (2,f(2)) STEP 2 Now we have to calculate the limit. 3x1 x+2 x+13 3(x+1) 3(x+1) (x2) lim x2 = lim x2 = lim x2 = lim (x2)(x+1)3 x2 x2 x2 x2 = lim 3(x+1) = 3(2+1)= 91 x2 x+13 1 Thus lim x2 = 9 x2