Mastering Limits: A Polynomial Approach To Solving Math Problems

lim x->∞ pow top < pow bottom

0

To solve this limit, we must consider the behavior of the numerator and denominator as x approaches infinity.

If the highest power in the numerator is less than the highest power in the denominator, then the limit will approach zero. This is because the numerator will grow at a slower rate than the denominator and will eventually be negligible compared to the denominator.

So, if we have:

lim x->∞ (x^2 + 3x + 1) / (x^3 + x^2 + 1)

we can see that the highest power in the numerator is x^2, while the highest power in the denominator is x^3. Therefore, as x approaches infinity, the denominator will grow much faster than the numerator, and the limit will approach zero:

lim x->∞ (x^2 + 3x + 1) / (x^3 + x^2 + 1) = 0

Note that we don’t need to actually calculate the limit to determine its value. We can use our understanding of the behavior of polynomials to determine the answer.

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