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The purpose of investigating each limit in "Investigate Each Limit....(B)" is to determine the behavior of a function as its input approaches a specific value. This can help us understand the overall behavior of the function and make predictions about its values.
Some common techniques used to investigate limits include algebraic manipulation, substitution, factoring, and the use of limit laws such as the sum, difference, and product rules.
A limit exists if the left-hand limit and right-hand limit are equal at the given value of x. If they are not equal, the limit does not exist. Another way to determine if a limit exists is to graph the function and see if there is a hole or a vertical asymptote at the given value of x.
A one-sided limit only considers the behavior of a function as it approaches a specific value from one direction (either the left or right). A two-sided limit considers the behavior of a function as it approaches a specific value from both the left and right directions.
Investigating limits can help us make predictions about the behavior of a system or process. For example, in physics, we can use limits to determine the velocity or acceleration of an object at a specific time. In economics, we can use limits to analyze the demand and supply of a product. In engineering, we can use limits to optimize the performance of a system.