Summary
Highlights
A key theorem states that for a positive rational number 'k' and any real number 'c', the limit of c/x^k as x approaches infinity (or negative infinity) is always zero. This is fundamental for evaluating limits of rational functions at infinity. When dealing with rational functions as x approaches infinity or negative infinity, the standard procedure is to divide both the numerator and denominator by the highest power of x found in the denominator.
Several examples illustrate this method. For lim (2x^2 - 5) / (3x^2 + x + 2) as x approaches negative infinity, dividing by x^2 yields 2/3. For lim (2x^2 - 5) / (3x^4 + x + 2) as x approaches infinity, dividing by x^4 results in 0/3 = 0. In cases where the highest power in the numerator is greater than in the denominator, such as lim (2x^3 - 5) / (3x^2 + x + 2) as x approaches infinity, the limit will be infinity.
When functions involve square roots, special care is needed. For f(x) = sqrt(9x^2 + 2) / (4x + 3), evaluating the limit as x approaches positive infinity requires factoring x^2 from under the square root, which comes out as x (since x is positive). The limit then simplifies to 3/4. However, if x approaches negative infinity, sqrt(x^2) becomes |x|, which is -x (since x is negative). This results in a limit of -3/4. This distinction is crucial for getting the correct sign.
The section, titled 'Limits Involving Infinity,' begins by examining the function f(x) = 1/(x-2). Its graph is similar to 1/x but shifted two units to the right. The dashed line at x=2 is a vertical asymptote, which the graph never crosses but approaches. When x approaches 2 from the right, the function tends to positive infinity, and when x approaches 2 from the left, it tends to negative infinity. This illustrates that if direct substitution yields a non-zero number over zero, one must evaluate the limit from both the right and left sides to determine if it is positive or negative infinity.
A vertical asymptote exists at x = a if the limit of the function as x approaches 'a' (from either side or both) results in positive or negative infinity. Examples are provided where approaching 'a' from the left or right can lead to positive or negative infinity, establishing 'a' as a vertical asymptote. It's crucial that the function approaches infinity from at least one side for a vertical asymptote to be present.
The lecture demonstrates how to find the limit of 1/(x-4)^3 as x approaches 4. Because direct substitution gives 1/0, it's necessary to evaluate from the left and right. Approaching from the left (e.g., using x=3) yields negative infinity, while approaching from the right (e.g., using x=5) yields positive infinity. Since the limits from both sides are not equal, the overall limit does not exist. The graph of this function confirms a vertical asymptote at x=4.
Unlike vertical asymptotes, where the graph cannot cross, horizontal asymptotes can be crossed by the function at some points, but the function's behavior at positive or negative infinity is what matters. The function f(x) = 2 + 1/x is used as an example, demonstrating that as x approaches positive or negative infinity, the function approaches y=2, making y=2 a horizontal asymptote. This is because 1/infinity or 1/-infinity both equal zero.