Investigating Limits: What Happens When x Approaches a Constant or Zero?

Thread starter nycmathguy
Start date Jun 17, 2021

In summary, investigating limits involves determining the behavior of a function as the input approaches a certain value, such as a constant or zero. This can be done through numerical evaluation or algebraic manipulation, and can provide insights into the behavior of a function at a specific point or as a whole. Approaching a constant can result in a finite limit, while approaching zero can lead to a variety of outcomes such as a limit that does not exist, a limit that approaches infinity, or a limit that approaches a finite value. Understanding limits is essential in calculus and other branches of mathematics.

Jun 17, 2021

nycmathguy

Homework Statement: Investigate each limit.

Relevant Equations: See attachment of piecewise function.

Investigate each limit.

1. lim f(x) x→3

2. lim f(x) x→0

See attachment.

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Jun 17, 2021

Mark44

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Thread closed. This is nearly identical to your other thread with almost the same name. When you figure this one out, you'll be able to figure this one out.

1. What is the purpose of investigating each limit in "Investigate Each Limit....(B)"?

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.

2. What are some common techniques used to investigate limits?

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.

3. How do we determine if a limit exists or not?

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.

4. What is the difference between a one-sided limit and a two-sided limit?

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.

5. How can investigating limits help us in real-world applications?

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.