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Angeline Ling
- 12
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Homework Statement
lim [(5^(n+1))+(7^(n+1))]/(5^n-7^n) as x→infinity
lim [sec x - tan x] as x → ∏/2
A limit in calculus is a fundamental concept that describes the behavior of a function as the input approaches a specific value. It is denoted by the symbol "lim" and is used to study the properties and continuity of functions.
The limit of a function can be found by evaluating the function at different values of the input that are approaching the specific value. If the function approaches a single value as the input gets closer to the specific value, then that single value is the limit of the function.
A one-sided limit only considers the behavior of the function as the input approaches the specific value from one direction (either from the left or from the right). A two-sided limit considers the behavior of the function from both directions and the limit only exists if the function approaches the same value from both directions.
Yes, a function can have a limit at a point but not be defined at that point. This means that the function approaches a specific value as the input approaches the point, but the function itself is not defined at that point.
Limits are used in real-world applications to solve problems involving rates of change, optimization, and approximations. For example, limits can be used to determine the maximum height of a projectile, the optimal production level for a company, or the slope of a curve at a specific point.