Evaluate using L'Hopital's rule:

Step-by-step explanation:

We are given:

And we want to evaluate it using L'Hopital's Rule.

First, using direct substitution, we will acquire:

Which is indeterminate.

In order to apply L'Hopital's Rule, we first need to manipulate the expression. We will let:

By taking the natural log of both sides:

And by taking the limit as x approaches 1 from the right of both functions:

Rewrite:

Using direct substitution on the right will result in 0/0. Hence, we can now apply L'Hopital's Rule:

Simplify:

Simplify:

Now, by using direct substitution, we will acquire:

Hence, we will apply L'Hopital's Rule once more. Utilize the product rule:

Finally, direct substitution yields:

Thus:

By the Composite Function Property for limits:

Raising both sides to e produces:

Therefore:

Substitution:

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation:

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step-by-step explanation: