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 guidance

1. 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.
2. 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|.
3. 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|.
4. Calculate the absolute value: The absolute value of -7 is 7. Therefore, the limit of |ƒ(t)| as t approaches t₀ is 7.
5. 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.

Key Concepts

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.

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.

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.