Questions: Evaluate the limit using L'Hôpital's rule
[
lim x rightarrow 0 frace^x+x-19 x
]
Transcript text: Evaluate the limit using L'Hôpital's rule
\[
\lim _{x \rightarrow 0} \frac{e^{x}+x-1}{9 x}
\]
$\square$
Solution
Solution Steps
To evaluate the limit using L'Hôpital's rule, first check if the limit is in an indeterminate form like 00 or ∞∞. If it is, differentiate the numerator and the denominator separately and then take the limit again. Repeat this process if necessary until the limit can be evaluated directly.
Step 1: Evaluate the Initial Limit
We start by evaluating the limit:
x→0lim9xex+x−1
Substituting x=0 gives us:
9⋅0e0+0−1=01+0−1=00
This is an indeterminate form, so we can apply L'Hôpital's rule.
Step 2: Differentiate the Numerator and Denominator
We differentiate the numerator and the denominator:
The derivative of the numerator ex+x−1 is ex+1.
The derivative of the denominator 9x is 9.
Step 3: Apply L'Hôpital's Rule
Now we apply L'Hôpital's rule:
x→0lim9ex+1
Substituting x=0 gives us:
9e0+1=91+1=92