lim x->∞ pow top = pow bottom
coefficients
In the given limit, we have a function in which the numerator and denominator both have some power functions. To solve this limit, we need to simplify the expression in such a way that it becomes easier to evaluate the limit.
One way to simplify the expression is to divide both the numerator and denominator by the highest power of x in the expression. This process is called dividing through by the highest power of x.
So, let’s divide both numerator and denominator by x^m, where m is the highest power of x in the expression. In this case, m can be found by comparing the degrees of the numerator and denominator.
Let’s assume that the degree of the numerator is n and the degree of the denominator is p.
If n > p, then m = n. If p > n, then m = p. If n = p, then m = n = p.
After dividing through by x^m, we get:
lim x->∞ (pow top/x^m) / (pow bottom/x^m)
Now, we can simplify the expression further by using the properties of exponents. The division of two powers with the same base is equal to subtracting the exponents, so we have:
lim x->∞ pow(top – bottom) / x^m
Since the denominator is a power function that approaches infinity as x approaches infinity, the only factor that determines the limit is the numerator.
If top – bottom > 0, then the numerator approaches infinity as x approaches infinity, and the limit is positive infinity.
If top – bottom < 0, then the numerator approaches negative infinity as x approaches infinity, and the limit is negative infinity.
If top - bottom = 0, then the numerator approaches zero as x approaches infinity, and the limit is zero.
Therefore, the limit of the given expression as x approaches infinity is:
- Positive infinity if top - bottom > 0
– Negative infinity if top – bottom < 0
- Zero if top - bottom = 0
Thus, we have solved the given limit.
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