교과서 질문
Determine the following limits.
a. lim x→−2^+ (x − 4) / x(x + 2)
352
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Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
2장, 문제 26b
Determine the following limits.
b. lim x→−2^− (x − 4) / x(x + 2)
검증된 단계별 안내1
Step 1: Identify the type of limit. Since the limit is as x approaches -2 from the left (x \(\to\) -2^-), we need to consider the behavior of the function as x gets closer to -2 from values less than -2.
Step 2: Analyze the function \( \frac{x - 4}{x(x + 2)} \). Notice that the denominator becomes zero when x = -2, which suggests a potential vertical asymptote at x = -2.
Step 3: Determine the sign of the function as x approaches -2 from the left. For x slightly less than -2, both x and (x + 2) are negative, making the denominator positive. The numerator (x - 4) is negative, resulting in the entire fraction being negative.
Step 4: Consider the magnitude of the function as x approaches -2 from the left. As x gets closer to -2, the denominator approaches zero, causing the magnitude of the fraction to increase without bound.
Step 5: Conclude the behavior of the limit. Since the function becomes increasingly negative as x approaches -2 from the left, the limit is negative infinity.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
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Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit as x approaches -2 from the left, which is denoted as x→−2^−.
추천 영상:
05:50
One-Sided Limits
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only. The notation x→−2^− indicates that we are considering the limit as x approaches -2 from values less than -2. This is crucial for analyzing functions that may have different behaviors when approached from different directions.
추천 영상:
05:50One-Sided Limits
Rational Functions
A rational function is a function that can be expressed as the ratio of two polynomials. In this limit problem, the function (x − 4) / (x(x + 2)) is a rational function. Understanding how to simplify and evaluate limits involving rational functions is essential, especially when determining points of discontinuity or indeterminate forms.
추천 영상:
6:04Intro to Rational Functions
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