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Answer :
We have established that the limit of the equation as x approaches 0 is 5/2 using l'Hôpital's Rule. Let's start by differentiating the numerator and the denominator separately.
To solve this limit using l'Hôpital's Rule, we need to rewrite the given limit in a form where the numerator and denominator approach zero or infinity.
1. Differentiate the numerator:
The numerator is [tex]x^2[/tex] + 5x. Taking the derivative, we get 2x + 5.
2. Differentiate the denominator:
The denominator is [tex]e^(2x)[/tex] - 1. Taking the derivative, we get 2[tex]e^(2x)[/tex].
3. Rewrite the limit:
Now we can rewrite the given limit using l'Hôpital's Rule. It becomes:
lim(x→0) (2x + 5) / (2[tex]e^(2x)[/tex])
4. Evaluate the limit:
Substituting x = 0 into the expression, we get:
(2(0) + 5) / (2[tex]e^(2(0))[/tex]) = 5 / 2
Conclusion:
Using l'Hôpital's Rule, we have determined that the limit of the given expression as x approaches 0 is 5/2.
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