How To Find The Limit At Infinity

The video starts by explaining how to find the limit of a function as x approaches infinity. For example, for x squared, as x becomes very large, x squared also becomes infinity. Similarly, as x approaches negative infinity, x squared results in positive infinity.

For polynomial functions, when finding limits at infinity, only the term with the highest degree is significant. For instance, in 5 + 2x - x cubed, the limit as x approaches negative infinity is determined solely by -x cubed, leading to positive infinity. Another example analyzes -5x to the fourth as x approaches negative infinity, resulting in negative infinity.

The video then covers rational functions. It explains that for 1/x, as x approaches infinity, the limit is zero because the denominator grows infinitely large. This concept applies generally: if the degree of the denominator is greater than the numerator (bottom heavy), the limit as x approaches infinity is always zero. This is demonstrated using 5x + 2 / 7x - x squared.

When the degree of the numerator and denominator are the same, the limit as x approaches infinity is the ratio of their leading coefficients. For example, the limit of (8x squared - 5x) / (4x squared + 7) as x approaches infinity is 8/4, which simplifies to 2. This is shown by dividing both numerator and denominator by the highest power of x.

If the degree of the numerator is greater than the denominator (top heavy), the insignificant terms can be removed to simplify the expression. For (5x + 6x squared) / (3x - 8) as x approaches infinity, it simplifies to (6x squared) / (3x) or 2x, leading to positive infinity. Another example, with 5 + 2x - 3x cubed / 4x squared + 9x - 7, also demonstrates this simplification.