Question

Use a calculator to estimate the values of the following limits to two decimal places.

lim
h → 0 (2.7^h − 1)/h

lim
h → 0 (2.8^h − 1)/h

Answer 1

you don't need to use a calculator for this problem, you need to do is differentiate with the top and the bottom with respect to h

Answer 2

see at your level anything divided by zero is equal to infinite, so you use a rule called the "L'Hospital's rule", which indicates that you may differentiate the numerator and the denominator

Answer 3

you can also use in your differentiation the D operator if you kindly know what it is?

Answer 4

so the answer to the first part would be 1.0296... and for 2 decimal places 1.02

Answer 5

can you work out the other one yourself, or if you need my help i will do

Answer 6

ok let me prove it for you

Answer 7

recall \[a ^{x}, f' = a ^{x}\ln(a)\]

Answer 8

now in your case because a=2.8 and the limit ->0, so all you need to do is to use your calculator to take the natural log of 2.8 or 2.7

Answer 9

because anything to the power of zero is one

Answer 10

for 2.7 limit the answer would be 0.99

Answer 11

i hope that makes sense to you