Determine the following limits. lim x→∞ $$6x2/(4x^2$$+√(16x4 + x2))

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Briggs 3rd Edition

Ch. 2 - Limits

문제 32

2장, 문제 32

Determine the following limits.

lim x→∞ 6x²/(4x^2+√(16x⁴ + x²))

검증된 단계별 안내

Identify the dominant terms in the numerator and the denominator. In the numerator, the dominant term is $6x^2$. In the denominator, the dominant term is $4x^2$ and the term under the square root is $16x^4$.

Factor out the highest power of $x$ from both the numerator and the denominator. In this case, factor $x^2$ from both.

Rewrite the expression: $\frac{6x^2}{4x^2 + \sqrt{16x^4 + x^2}} = \frac{6}{4 + \sqrt{16 + \frac{1}{x^2}}}$.

As $x$ approaches infinity, the term $\frac{1}{x^2}$ approaches zero. Simplify the expression to $\frac{6}{4 + \sqrt{16}}$.

Calculate the simplified expression by evaluating the square root and performing the arithmetic operations.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the variable approaches infinity. This concept is crucial for understanding how functions behave for very large values, often simplifying expressions by focusing on the highest degree terms in polynomials or rational functions.

Dominant Terms

In the context of limits, dominant terms are the terms in a polynomial or rational expression that have the highest degree and thus dictate the behavior of the function as the variable approaches infinity. Identifying these terms allows for simplification of the limit calculation, as lower degree terms become negligible.

Rational Functions

A rational function is a ratio of two polynomials. Understanding the properties of rational functions, including their limits, is essential for solving limit problems. The behavior of rational functions at infinity often depends on the degrees of the numerator and denominator, which can lead to finite limits, infinite limits, or limits that do not exist.

Determine the following limits. lim x→∞ 6x2/(4x^2+√(16x4 + x2))