Question

Finding Limits and Relative Extrema Use spreadsheet to complete the table using

f(x) = In x/x.
x 1 5 10 10^2 10^4 10^6
f(x)
(a) Use the table to estimate the limit: lim f(x).
(b) Use a graphing utility to estimate the relative extrema of f.

Answer 1

a) 0 b) Maximum (2<x<3, 0<y<1)

Explanation:

a) Through the table we can estimate the value of the limit as 0, since it starts with 0 goes up and then goes down to 0

Verifying:

`\lim_(x\rightarrow \infty)(ln(x))/(x)\Rightarrow \lim_(x\rightarrow \infty)\frac{\frac{\mathrm{d} }{\mathrm{d} x}[ln(x)]}{\frac{\mathrm{d} }{\mathrm{d} x}[x]}\Rightarrow \lim_(x\rightarrow \infty)((1)/(x))/(1)\Rightarrow (\lim_(x\rightarrow \infty)1)/(\lim_(x\rightarrow \infty)x)=0`

b) The Relative extrema was estimated here using Geogebra. Estimating it considering [0,5] we could say: (2<x<3, 0<y<1). Calculating it using Geogebra applet:

`(e,(1)/(e))=(2.72,0.36)`