Guide To Evaluating Limits Of Functions Using Exponents At Infinity For Positive Integers A And B

lim x->∞ pow top > pow bottom

∞, -∞

To evaluate this limit, we need to consider the behavior of the top and bottom functions as x approaches infinity. If the numerator (top) grows at a faster rate than the denominator (bottom), the overall fraction will approach infinity. If the denominator grows at a faster rate than the numerator, the overall fraction will approach zero. If both the top and bottom grow at the same rate, then we need to evaluate further.

Assuming pow means power, we can rewrite the expression as:

lim x->∞ (top)^a / (bottom)^b

where a and b are positive integers. We can apply the property of exponents that says (x^a)^b = x^(ab) to rewrite this as:

lim x->∞ (top/bottom)^(a/b)

So we need to consider the behavior of the top/bottom fraction as x approaches infinity. If the fraction approaches a positive number, then the limit is a positive number raised to a power, which is also a positive number. If the fraction approaches infinity, then the limit is infinity raised to a power, which is still infinity. If the fraction approaches zero, then the limit is zero raised to a power, which is zero.

Therefore, the limit will be:

– infinity, if top/bottom -> infinity
– 0, if top/bottom -> 0
– a positive number, if top/bottom -> a positive number

Note that if the exponent a/b is a fraction like 1/2 or 2/3, then we can still use the same reasoning – we just need to take the square root or cube root, respectively, of the top/bottom fraction before raising it to the power.

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