Applying L’Hopital’S Rule: How To Solve Indeterminate Limits With Derivatives

lim x->∞ pow top = pow bottom

coefficients

In order to solve this limit problem, we can use L’Hopital’s Rule, which states that if we have an indeterminate form of the type ∞/∞ or 0/0, we can take the derivative of the top and bottom of the fraction and evaluate the limit again until we get a non-indeterminate form.

To apply L’Hopital’s Rule to this limit problem, we need to find the derivative of both the top and bottom of the fraction. Let’s start by finding the derivative of the top:

d/dx(pow top) = lim x->∞ [(pow top) * ln(pow top)]

Next, let’s find the derivative of the bottom:

d/dx(pow bottom) = lim x->∞ [(pow bottom) * ln(pow bottom)]

Now we can rewrite the original limit problem as:

lim x->∞ [(pow top) * ln(pow top)] / [(pow bottom) * ln(pow bottom)]

Since we still have an indeterminate form of ∞/∞, we need to apply L’Hopital’s Rule again by taking the derivative of both the top and bottom of the fraction:

d/dx[(pow top) * ln(pow top)] = lim x->∞ [(pow top) * (ln(pow top) + 1)]

d/dx[(pow bottom) * ln(pow bottom)] = lim x->∞ [(pow bottom) * (ln(pow bottom) + 1)]

Now we can rewrite the limit problem again as:

lim x->∞ [(pow top) * (ln(pow top) + 1)] / [(pow bottom) * (ln(pow bottom) + 1)]

Since we still have an indeterminate form of ∞/∞, we need to apply L’Hopital’s Rule one more time by taking the derivative of both the top and bottom of the fraction:

d/dx[(pow top) * (ln(pow top) + 1)] = lim x->∞ [pow top / x]

d/dx[(pow bottom) * (ln(pow bottom) + 1)] = lim x->∞ [pow bottom / x]

Now we can rewrite the limit problem once more as:

lim x->∞ [pow top / x] / [pow bottom / x]

At this point, we can simplify the fraction by dividing both the top and bottom by x:

lim x->∞ pow top / pow bottom

Since we have taken the limit to infinity and both the numerator and denominator are powers of x, the limit evaluates to zero if the degree of the numerator is less than the degree of the denominator, while it diverges to either infinity or negative infinity if the degree of the numerator is greater than or equal to the degree of the denominator.

Therefore, the limit in question, lim x->∞ pow top = pow bottom, cannot be evaluated without knowing the specific values of pow top and pow bottom, as the degree of the powers in the numerator and denominator may determine the behavior of this limit.

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