Question

Find the indicated limit. Note that L'Hopital's rule may not apply or may require multiple applications.

lim e^9x -1-9x/x^2
x→0

Answer 1

40.5

Explanation:

Given the limit of the function expressed as:

`\lim_(x \to 0) (e^(9x)-1-9x)/(x^2) \\`

Step 1: Substitute x = 0 into the function:

`\lim_(x \to 0) (e^(9x)-1-9x)/(x^2) \\= (e^(9(0))-1-9(0))/((0)^2)\\= \frac{{1-1-0}}{0}\\= (0)/(0) (ind)`

Step 2: Apply L'hopital rule

`\lim_(x \to 0) (d/dx(e^(9x)-1-9x))/(d/dx(x^2)) \\= \lim_(x \to 0) ((9e^(9x)-9))/((2x))\\= ((9e^(9(0))-9))/((2(0)))\\= (9-9)/(0) \\= (0)/(0) (ind)`

Step 3: Apply L'hopital rule again

`\lim_(x \to 0) (d/dx(9e^(9x)-9))/(d/dx(2x))\\ = \lim_(x \to 0) (81e^(9x))/(2) \\=(81e^(0))/((2))\\= (81)/(2) \\= 40.5`

Hence the limit of the function is 40.5