Question

Use the definition of e, and numerical approximations as y approaches 0 from the left, and y approaches 0 from the right.

(b) Use a calculator to estimate the values of the limits given below, correct to two decimal places.
lim (y -> 0) (2.7[sup]y[/sup] - 1)/ y and lim (y -> 0) (2.8[sup]y[/sup] - 1)/ y

____ < 1 < _______

What can you conclude about the value of e?

________ < e < _________

Answer 1

Question #1838 Use the definition of $e$, and numerical approximations as y approaches 0 from the left, and y approaches 0 from the right.

(b) Use a calculator to estimate the values of the limits given below, correct to two decimal places. $\lim_{y\to 0}(2.7^y -1) / y$ and $\lim_{y\to 0}(2.8^y -1) / y$ What can you conclude about the value of e?

Solution. The first question is unclear. For the next use if $a > 0$ then $\lim_{x\to 0}\frac{a^x - 1}{x} = \ln a$. Hence $\lim_{y\to 0}(2.7^y -1) / y = \ln 2.7\approx 0.99$ and $\lim_{y\to 0}(2.8^y -1) / y = \log 2.8\approx 1.03$. Hence $2.7 < e < 2.8$.