Active Calculus - Gvsu - Grand Valley State University

2.8. USING DERIVATIVES TO EVALUATE LIMITS 612.8 Using Derivatives to Evaluate LimitsPreview Activity 2.8. Let h be the function given by h(x) = x5 + x − 2x 2 .− 1(a) What is the domain of h?x 5 + x − 2(b) Explain why limx→1 x 2 results in an indeterminate form.− 1(c) Next we will investigate the behavior of both the numerator and denominator of h nearthe point where x = 1. Let f(x) = x 5 +x−2 and g(x) = x 2 −1. Find the local linearizationsof f and g at a = 1, and call these functions L f (x) and L g (x), respectively.(d) Explain why h(x) ≈ L f (x)for x near a = 1.L g (x)(e) Using your work from (c), evaluateL f (x)limx→1 L g (x) .What do you think your result tells us about limx→1h(x)?(f) Investigate the function h(x) graphically and numerically near x = 1. What do you thinkis the value of limx→1h(x)?⊲⊳