In calculus, understanding limits and continuity is crucial for solving complex mathematical problems. Homework assignments on this topic help students practice and reinforce their knowledge. In this article, we will provide comprehensive answers to various questions related to 1.6 Limits and Continuity, ensuring a thorough understanding of the concepts involved.
Solutions to 1.6 Limits and Continuity Homework Questions
Question: Find the limit of f(x) as x approaches 2, given that f(x) = (x^2 + 3x – 4) / (x – 2).
Answer: To find the limit, we need to simplify the expression by canceling out the common factors. By factoring the numerator, we get (x – 1)(x + 4). As x approaches 2, we can substitute this value into the simplified expression, resulting in (2 – 1)(2 + 4) = 6. Thus, the limit of f(x) as x approaches 2 is 6.
Question: Determine the limit of g(x) as x approaches -3, given that g(x) = (x^3 + 27) / (x + 3).
Answer: To solve this question, we can factor the numerator using the sum of cubes formula, which gives us (x + 3)(x^2 – 3x + 9). By canceling out the common factor (x + 3), we are left with (x^2 – 3x + 9). Substituting x = -3 into the simplified expression, we get (-3)^2 – 3(-3) + 9 = 9 + 9 + 9 = 27. Hence, the limit of g(x) as x approaches -3 is 27.
Comprehensive Answers for 1.6 Limits and Continuity Assignments
Question: Determine the limit of h(x) as x approaches 0, given that h(x) = (sin3x)/(5x).
Answer: In this case, we can use the Squeeze Theorem to find the limit. As x approaches 0, the value of sin3x will approach 0, and the denominator 5x will also approach 0. By applying the Squeeze Theorem, we can conclude that the limit of h(x) as x approaches 0 is 0.
Question: Find the limit of k(x) as x approaches infinity, given that k(x) = (3x^2 – 2x + 1) / (x^3 + 5).
Answer: To determine the limit as x approaches infinity, we need to consider the degrees of the numerator and denominator. Since the degree of the numerator is 2 and the degree of the denominator is 3, we can conclude that the limit of k(x) as x approaches infinity is 0. Dividing the highest power term in the numerator by the highest power term in the denominator yields 0, indicating that the numerator grows at a slower rate than the denominator as x approaches infinity.
By providing solutions and comprehensive answers to various questions on 1.6 Limits and Continuity, this article aimed to enhance understanding and mastery of the topic. Remember, limits and continuity are fundamental concepts in calculus, and continued practice helps solidify these concepts. By familiarizing yourself with different problem-solving techniques and regularly reviewing these concepts, you can confidently tackle more advanced calculus problems in the future.