Textbook Question
Horizontal and Vertical Asymptotes
Use limits to determine the equations for all horizontal asymptotes.
_____
√x² + 4
c. g(x) = -----------
x
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1. Limits and Continuity
Finding Limits Algebraically
1:54 minutesProblem 3f
Textbook Question
Limits and Continuity
Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.
f. | ƒ(t) |
Verified step by step guidance1
Understand the problem: We are given that the limit of ƒ(t) as t approaches t₀ is -7, and the limit of g(t) as t approaches t₀ is 0. We need to find the limit of |ƒ(t)| as t approaches t₀.
Recall the property of limits: If the limit of a function exists as t approaches a certain point, then the limit of the absolute value of that function also exists. Specifically, if lim t → t₀ ƒ(t) = L, then lim t → t₀ |ƒ(t)| = |L|.
Apply the property to the given function: Since we know that lim t → t₀ ƒ(t) = -7, we can use the property to find that lim t → t₀ |ƒ(t)| = |-7|.
Calculate the absolute value: The absolute value of -7 is 7. Therefore, the limit of |ƒ(t)| as t approaches t₀ is 7.
Conclude the solution: By applying the limit property for absolute values, we have determined that the limit of |ƒ(t)| as t approaches t₀ is 7.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit describes the value that a function approaches as the input approaches a certain point. In this context, the limit of ƒ(t) as t approaches t₀ is given as -7, indicating that as t gets closer to t₀, ƒ(t) gets closer to -7. Understanding limits is crucial for analyzing the behavior of functions near specific points.
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Absolute Value Function
The absolute value function, denoted as |ƒ(t)|, transforms any real number into its non-negative counterpart. This means that if ƒ(t) approaches -7, then |ƒ(t)| will approach 7 as t approaches t₀. Recognizing how the absolute value affects limits is essential for solving the given problem.
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Continuity
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. In this case, since the limit of ƒ(t) exists and is finite, we can infer that the limit of |ƒ(t)| as t approaches t₀ will also exist and be equal to 7, demonstrating the continuity of the absolute value function at that limit.
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