[Calc] Limit as x approaches INFINITY (but there is no horizontal asymptote)

Описание: This video focuses on understanding and calculating limits as x approaches infinity, particularly their application in identifying horizontal asymptotes of functions. It also addresses scenarios where no horizontal asymptote exists because the numerator's degree is greater than the denominator's. The video further covers a critical, often-tested scenario involving limits as x approaches negative infinity, especially when square roots are present.

The objective of the video is to teach viewers how to effectively determine horizontal asymptotes, or confirm their absence, using two distinct yet complementary methods:

• Conceptual Approach: This method teaches viewers to quickly analyze rational functions and functions with square roots by considering x as a "gigantic number," whether positive or negative. It emphasizes identifying the fastest-growing terms (highest powers of x) and disregarding minor constant terms to rapidly determine the limit.

For cases where the numerator's degree is higher, this approach helps understand that the numerator will "outgrow" the denominator, leading to a limit of infinity and thus no horizontal asymptote. For limits approaching negative infinity with square roots, it's crucial to conceptually understand that sqrt(x^2) represents |x|, which simplifies to -x when x is negative, guiding the correct sign of the limit. This approach is particularly useful for multiple-choice questions.

• Algebraic Approach: This is the formal method required for free-response questions and addresses the indeterminate infinity/infinity form. Key techniques include:
◦ Dividing every term in the numerator and denominator by the greatest power of x.
◦ A crucial situational thinking rule for when the numerator's degree is higher than the denominator's: to algebraically demonstrate that the limit is infinity (i.e., no horizontal asymptote), you should divide by the highest power of the denominator (e.g., x), rather than the highest power of the numerator. This prevents the denominator from becoming zero after substitution and clearly shows the limit approaching infinity.
◦ Correctly handling square roots in the denominator or numerator by recognizing that x outside a square root is equivalent to sqrt(x^2) inside.
◦ For limits approaching negative infinity with square roots, the video highlights an essential analytical step: when dividing terms by x, it's vital to acknowledge that x is a negative value. Therefore, to maintain the correct sign of the original expression, a manual negative sign must be introduced when converting x into sqrt(x^2) inside the square root (e.g., sqrt(x^2 + 1) / x becomes -sqrt((x^2 + 1)/x^2)).
◦ Applying the property that terms like 1/x or 1/x^2 approach 0 as x approaches infinity (or negative infinity).

The video stresses the importance of mastering both conceptual and algebraic approaches, with particular emphasis on the analytical step for negative infinity and square roots, as this specific scenario is frequently tested due to its common pitfalls. It also gives a heads-up about slant asymptotes, noting that when the numerator's degree is greater than the denominator's, long or synthetic division might be used in future topics to re-express the fraction as a polynomial, indicating a slant asymptote.

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