# Infinite Limit: Solving \frac{x}{x-5}

• hockeyfghts5
In summary, the problem involves finding the limit of the expression (x/(x-5)) as x approaches infinity. The solution involves realizing that as x gets larger and larger, the expression (1/(1-5/x)) gets closer and closer to 1. Therefore, the limit of the original expression is 1.
hockeyfghts5

## Homework Statement

$$\frac{x}{x-5}$$

$$lim x\rightarrow\infty$$2. The attempt at a solution$$\frac{x}{x-5} \equiv \frac{1}{1-\frac{5}{x}}$$

$$lim x\rightarrow\infty$$

how do i know what to plug into x to solve the problem. by the way it equals 1

You don't have to plug anything into x. As x gets larger and larger, what does 1/(1 - 5/x) approach? Can you convince yourself that this expression gets closer and closer to 1 the larger x gets?

so would you just plug zero for x and the simplify the problem which would give 1?

Perhaps to elaborate slightly on what Mark44 has posted: At this moment in time consider lim (x -> infinity) 1/x. We want to find what 1/x approaches as x becomes arbitrarily large. Let us first consider x = 10, then our expression becomes 1/10 = 0.1. Now suppose x = 100, then our expression becomes 1/100 = 0.01. Now suppose x = 100000, than our expression becomes 1/100000 = 0.00001, and for even larger x the expression becomes even smaller; hence, the limit aproaches 0.

Edit: You would not let x = 0, you may let the expression 5/x approach zero as x tends to inifinity though.

Thanks, that makes sense.

## 1. What is an infinite limit?

An infinite limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value. In this case, it refers to the behavior of the function $\frac\left\{x\right\}\left\{x-5\right\}$ as x approaches 5.

## 2. How do you solve for an infinite limit?

To solve for an infinite limit, you need to algebraically manipulate the function to simplify it and then substitute the approaching value into the simplified function. In this case, you would simplify $\frac\left\{x\right\}\left\{x-5\right\}$ to $\frac\left\{1\right\}\left\{1-\frac\left\{5\right\}\left\{x\right\}\right\}$ and then substitute in 5 to get the answer of infinity.

## 3. What is the significance of an infinite limit?

An infinite limit can indicate that the function is approaching a vertical asymptote or that it has a point of discontinuity at the approaching value. In the context of this function, it means that as x gets closer and closer to 5, the function grows without bound.

## 4. Can an infinite limit ever equal a finite number?

No, an infinite limit will always equal either positive or negative infinity. This is because the function is growing without bound as the input approaches the given value.

## 5. How is an infinite limit different from a finite limit?

A finite limit is a limit that has a defined value, while an infinite limit does not. Additionally, a finite limit means that the function approaches a specific value as the input approaches a certain value, while an infinite limit means that the function grows without bound as the input approaches a certain value.

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