lim x->∞ pow top > pow bottom
∞, -∞
To evaluate the limit as x approaches infinity of pow top divided by pow bottom, where pow top and pow bottom are functions of x, we need to use the concept of limits at infinity.
If the degree of the numerator (pow top) is greater than the degree of the denominator (pow bottom), the limit as x approaches infinity will be infinity. On the other hand, if the degree of the denominator is greater than the degree of the numerator, the limit as x approaches infinity will be zero.
To formally prove this, let’s assume that pow top has a degree n (i.e., the highest power of x in pow top is x^n) and the degree of pow bottom is m (i.e., the highest power of x in pow bottom is x^m).
Case 1: n > m
If the degree of pow top is greater than the degree of pow bottom, we can simplify the expression by dividing every term of pow top by x^n, and every term of pow bottom by x^n. This will give us:
lim x->∞ ((a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0) / x^n) / ((b_m x^m + b_{m-1} x^{m-1} + … + b_1 x + b_0) / x^n)
Simplifying this expression, we get:
lim x->∞ (a_n + a_{n-1} (1/x) + … + a_1 (1/x)^{n-1} + a_0 (1/x)^n) / (b_m + b_{m-1} (1/x) + … + b_1 (1/x)^{m-1} + b_0 (1/x)^m)
As x approaches infinity, every term with a (1/x) factor will approach zero, leaving us with:
lim x->∞ a_n / b_m
Since n > m, the limit of pow top divided by pow bottom as x approaches infinity will be infinity because the highest power of x dominates the expression.
Case 2: n < m If the degree of pow bottom is greater than the degree of pow top, once again we can simplify the expression by dividing every term of pow top by x^m, and every term of pow bottom by x^m. This will give us: lim x->∞ ((a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0) / x^m) / ((b_m x^m + b_{m-1} x^{m-1} + … + b_1 x + b_0) / x^m)
Simplifying this expression, we get:
lim x->∞ (a_n (1/x)^{m-n} + a_{n-1} (1/x)^{m-n+1} + … + a_1 (1/x)^{m-1} + a_0 (1/x)^m) / (b_m + b_{m-1} (1/x) + … + b_1 (1/x)^{m-1} + b_0 (1/x)^m)
As x approaches infinity, every term with (1/x) factor will approach zero, leaving us with:
lim x->∞ 0 / b_m
Since m > n, this limit will equal zero because the highest power of x in the denominator dominates the expression.
In summary, if the degree of the numerator is greater than the degree of the denominator, the limit as x approaches infinity will be infinity. If the degree of the denominator is greater than the degree of the numerator, the limit as x approaches infinity will be zero.
More Answers:
Unlocking The Power Of Functions: The Intermediate Value Theorem (Ivt)The Three Types Of Discontinuities In Mathematical Functions: Removable, Jump, And Infinite.
Continuity In Math: Types And Importance For Calculus And Beyond