Limits with the precise definition of a limit

In summary: So let's start by assuming that the function is bounded and try to find a precise definition for a limit.
  • #1
newschoolgg
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0
Member warned about not including a solution effort

Homework Statement


Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement.
(a) lim x-> a [f(x) + g(x)] = infinity
(b) lim x-> a [f(x)g(x)] = infinity if c > 0
(c) lim x-> a [f(x)g(x)] = negative infinity if c < 0

Homework Equations


The limit laws would seem relevant, but the f(x) limit goes to infinity.

The Attempt at a Solution


I'm completely lost on how to start this problem. How would I prove something like the limit laws using the precise definition of a limit when one of the limits don't exist?
 
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  • #2
When you say that
$$\lim_{x \to a} f(x) = \infty,$$ it's not exactly the same as saying the limit doesn't exist. For example, the limit
$$\lim_{x \to 0} \sin \frac{1}{x}$$ doesn't exist, but you wouldn't say it's equal to infinity either. So what precisely does it mean when you write a limit equals infinity?
 
  • #3
newschoolgg said:

Homework Statement


Suppose that limit x-> a f(x)= infinity and limit x-> a g(x) = c, where c is a real number. Prove each statement.
(a) lim x-> a [f(x) + g(x)] = infinity
(b) lim x-> a [f(x)g(x)] = infinity if c > 0
(c) lim x-> a [f(x)g(x)] = negative infinity if c < 0

Homework Equations


The limit laws would seem relevant, but the f(x) limit goes to infinity.

The Attempt at a Solution


I'm completely lost on how to start this problem. How would I prove something like the limit laws using the precise definition of a limit when one of the limits don't exist?
Your textbook should have the precise definition of a limit when the function is unbounded. It doesn't use ##\delta## and ##\epsilon## as the normal limit does, but instead uses M and ##\delta##, where M is a large number.
 

FAQ: Limits with the precise definition of a limit

1. What is the precise definition of a limit?

The precise definition of a limit is the mathematical concept that describes the behavior of a function as the input approaches a specific value. It states that the limit of a function at a certain point exists if the values of the function approach a finite number as the input approaches that specific point.

2. Why is the precise definition of a limit important?

The precise definition of a limit is important because it allows us to rigorously analyze and understand the behavior of functions. It is the foundation of calculus and is used to solve various real-world problems in fields such as physics, engineering, and economics.

3. How is the precise definition of a limit different from the intuitive understanding of a limit?

The intuitive understanding of a limit is based on the idea that the values of a function get closer and closer to a certain value as the input approaches a specific point. The precise definition, on the other hand, involves using mathematical notation and strict criteria to determine the existence and value of a limit.

4. What are the three key components of the precise definition of a limit?

The three key components of the precise definition of a limit are the limit point (the specific value that the input approaches), the limit value (the finite number that the function approaches), and the difference between the limit point and input value (also known as the distance or delta).

5. How can the precise definition of a limit be used to determine if a function is continuous?

If the limit of a function at a certain point exists and is equal to the value of the function at that point, then the function is continuous at that point. This is because the precise definition of a limit ensures that the function approaches the same value from both the left and right sides of the limit point, indicating a smooth, unbroken graph.

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