교과서 질문
Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist.
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Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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 2.15
Determine the following limits.
lim x→1 x^3 − 7x^2 + 12x / 4 − x
검증된 단계별 안내1
Identify the limit expression: \( \lim_{{x \to 1}} \frac{x^3 - 7x^2 + 12x}{4 - x} \).
Check if direct substitution of \( x = 1 \) results in an indeterminate form. Substitute \( x = 1 \) into the numerator and denominator.
Since direct substitution results in an indeterminate form \( \frac{0}{0} \), apply algebraic manipulation to simplify the expression. Factor the numerator \( x^3 - 7x^2 + 12x \).
Factor out \( x \) from the numerator: \( x(x^2 - 7x + 12) \). Further factor \( x^2 - 7x + 12 \) into \( (x - 3)(x - 4) \).
Rewrite the expression as \( \lim_{{x \to 1}} \frac{x(x - 3)(x - 4)}{4 - x} \) and simplify by canceling common factors, then evaluate the limit.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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 specific points, which is crucial for evaluating functions that may not be defined at those points. In this case, we are interested in the limit as x approaches 1.
추천 영상:
05:50
One-Sided Limits
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This technique is often used to simplify expressions, especially when evaluating limits, as it can help eliminate indeterminate forms like 0/0. In the given limit, factoring the numerator will be essential to simplify the expression before substituting x = 1.
추천 영상:
6:04Introduction to Polynomial Functions
Indeterminate Forms
Indeterminate forms occur when direct substitution in a limit leads to an undefined expression, such as 0/0 or ∞/∞. Recognizing these forms is crucial because they indicate that further analysis, such as factoring or applying L'Hôpital's Rule, is needed to evaluate the limit correctly. In this problem, substituting x = 1 initially results in an indeterminate form.
추천 영상:
가이드 코스
3:56Slope-Intercept Form
관련 실천
교과서 질문
Evaluate each limit.
415
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교과서 질문
Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→3 x − 3 /|x − 3|
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교과서 질문
Evaluate each limit and justify your answer.
lim x→2 (3 / 2x^5−4x^2−50)^4
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교과서 질문
Use the definitions given in Exercise 57 to prove the following infinite limits.
lim x→1^+ 1 /1 − x=−∞
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교과서 질문
Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right?
f(x)=(2x−3)^2/3
252
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