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.

\(\lim _{h \rightarrow 0}\ \frac{(2+h)^4-16}{h}\)

SOLUTION STEP 1 Given is the slope of the curve y=f(x) Thus it is the derivative of the function f(x). (2+h) 16 ie,f(a) = lim h h0 Thus comparing with the definition of the derivative f(a+h)f(a) f(a) = lim h h0 4 4 Thus f(a) = 2 f(x) = x 4 Thus we can find that f(x) = x and a = 2 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. (2+h) 16 2 +4.2 h+6.2 h +4.2.h +h 16 lim = lim h0 h h0 h 16+32h+24h +8h +h 16 h +8h +24h +32h 3 2 lim h = lim h = lim h + 8h + 24h + 32 = 32 h0 h0 h0 (2+h) 16 Thus lim =32 h0 h