Why Does the Limit of \(x^2 - \frac{1}{x}\) as \(x\) Approaches 0 Not Exist?

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In summary, a limit problem is a mathematical concept used to determine the behavior of a function as the input approaches a specific value. It is important to solve these problems in various fields of science, such as physics, engineering, and economics, to understand the behavior of a function and make predictions about its output. There are different methods to solve a limit problem, and it is essential to avoid common mistakes, such as assuming a limit does not exist or forgetting to check for discontinuities. Limit problems also have real-life applications in business, economics, physics, engineering, and biology.
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carbz
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[SOLVED] Just one more limit problem.

Homework Statement


Find the limit


Homework Equations


[itex]\lim_{x \rightarrow 0} (x^2 - \frac{1}{x})[/itex]


The Attempt at a Solution


I got does not exist for this limit. This is since, when you break it down to two parts, the 1/x is undefined at 0.
 
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  • #2
That's correct.
 
  • #3
allright, thank you
 
  • #4
The fact that a function is undefined at a point does not imply the limit does not exist at that point.
 
  • #5
It is infinite on both sides of the point, it doesn't exist.
 

What is a limit problem?

A limit problem is a mathematical concept used to determine the behavior of a function as the input approaches a specific value. It helps to understand the trend or pattern of a function and its output as it approaches a certain point.

Why is it important to solve limit problems?

Solving limit problems is important in various fields of science, such as physics, engineering, and economics. It allows us to understand the behavior of a function and make predictions about its output. It also helps in finding the maximum or minimum values of a function, which is useful in optimization problems.

What are the different methods to solve a limit problem?

There are various methods to solve a limit problem, including direct substitution, factoring, rationalizing, and using L'Hôpital's rule. Each method is useful for different types of limit problems and can be chosen based on the complexity of the problem.

What are the common mistakes to avoid when solving a limit problem?

One common mistake is to assume that a limit does not exist if the function is undefined at that point. Another mistake is to forget to check for any discontinuities or holes in the function. It is also essential to simplify the expression before evaluating the limit to avoid errors.

How can limit problems be applied in real-life situations?

Limit problems have various real-life applications, such as determining the maximum or minimum values in business and economics, predicting the behavior of a system in physics and engineering, and understanding population growth and decay in biology. They also help in calculating derivatives, which are essential in many fields of science and engineering.

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