the graph of 𝑓(𝑥)=2𝑥−8𝑥−3.

Question

 the graph of 𝑓(𝑥)=2^x-8/x-3

Estimate the limit of 𝐿=lim𝑥→3𝑓(𝑥).

Answer 1

𝑓(𝑥) = (2^x – 8)/(x – 3)

Trying to calculate the limit L= lim_𝑥→3𝑓(𝑥) we run into a situation where both numerator and denominator approach zero. So we use L'Hôpital's rule, taking the derivative of numerator and denominator and calculating the resulting limit.

d/dx[2^x – 8] = 2^xln2 and d/dx[x – 3] = 1

So lim_x→3[(2^x-8)/(x-3)] = lim_x→3[(2^xln2)/1] = 2³ln2 = 8ln2

answer

L = lim_𝑥→3𝑓(𝑥) = 8ln2

Note on taking the derivative of 2^x

2^x = (e^ln2)^x = e^xln2

d/dx[2^x] = d/dx[e^xln2] = (e^xln2)ln2 (applying the chain rule)

Since e^xln2 = 2^x, we can substitute to obtain (e^xln2)ln2 = 2^xln2