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.
e. cos (g(t))

Verified step by step guidance

1. First, recall the limit property for composite functions: if lim(t → t₀) g(t) = L and the function h is continuous at L, then lim(t → t₀) h(g(t)) = h(L).
2. In this problem, we are given that lim(t → t₀) g(t) = 0. We need to determine if the function cos(x) is continuous at x = 0.
3. The cosine function, cos(x), is continuous for all real numbers, including at x = 0. Therefore, we can apply the limit property for composite functions.
4. Using the property, we substitute L = 0 into the continuous function cos(x), giving us lim(t → t₀) cos(g(t)) = cos(0).
5. Finally, evaluate cos(0) to find the limit. Remember, cos(0) is a well-known trigonometric value.

Key Concepts

Limits

A limit describes the value that a function approaches as the input approaches a certain point. In this context, we are interested in the behavior of the functions ƒ(t) and g(t) as t approaches t₀. Understanding limits is crucial for evaluating the continuity and behavior of functions at specific points.

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 lim t → t₀ ƒ(t) = -7, it suggests that ƒ(t) is continuous at t₀ if ƒ(t₀) = -7. Continuity is essential for ensuring that limits can be evaluated without abrupt changes in function values.

Composition of Functions

The composition of functions involves applying one function to the result of another. In this problem, we need to evaluate cos(g(t)) as t approaches t₀. Since we know the limit of g(t) as t approaches t₀ is 0, we can find the limit of the composition by substituting this limit into the outer function, cos(x), to determine the overall limit.