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Use l'Hôpital's rule to find the following limit. limx→[infinity]​7x3+45x3−3x​ limx→[infinity]​7x3+45x3−3x​= (Type an integer or a fraction

Answer :

The limit as x approaches infinity of the given function is infinity.

To apply l'Hôpital's Rule, we differentiate the numerator and the denominator separately until we no longer have an indeterminate form.

Given function:[tex]f(x) = 7x^3 + 45x^3 - 3x[/tex]

1. Take the derivative of the numerator:

[tex]f'(x) = 21x^2 + 135x^2 - 3[/tex]

2. Take the derivative of the denominator:

[tex]g'(x) = 1[/tex]

3. Apply l'Hôpital's Rule again to the new function f'(x) / g'(x):

[tex]lim (x- > ∞) f'(x) / g'(x) = lim (x- > ∞) (21x^2 + 135x^2 - 3) / 1[/tex]

4. Simplify the expression:

[tex]lim (x- > ∞) (21x^2 + 135x^2 - 3) = lim (x- > ∞) (156x^2 - 3)[/tex]

As x approaches infinity, the term with the highest power dominates. In this case, it is [tex]156x^2.[/tex]

5. Taking the limit:

[tex]lim (x- > infinity) (156x^2 - 3) = infinity[/tex]

Therefore, the limit as x approaches infinity of the given function is infinity.

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