# Solving a Limit Problem: Step-by-Step Guide

In summary, a limit problem is a mathematical concept used to find the value of a function at a specific point or as the input approaches a certain value. To solve a limit problem, one must evaluate the function, simplify it, and substitute the value to find the limit. There are different types of limit problems, including basic limits, limits at infinity, and limits involving various functions. Common mistakes to avoid when solving a limit problem include forgetting to check for discontinuities, incorrect algebraic rules, and not considering the function's behavior. Solving limit problems is important for understanding function behavior, solving real-world problems, and is a fundamental concept in calculus and other fields of mathematics and science.
Can someone please tell me how to solve a limit problem like this?

$$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$

This is my attempt to solve the problem:

$$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$
$$= \lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}} × \frac{\sqrt{x^2 + x} + \sqrt{x^2 - 3x}}{\sqrt{x^2 + x} + \sqrt{x^2 - 3x}}$$
$$= \lim_{x \to \infty} \frac{4(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{(x^2 + x) - (x^2 - 3x)}$$
$$= \lim_{x \to \infty} \frac{4(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{x^2 + x - x^2 + 3x}$$
$$= \lim_{x \to \infty} \frac{4(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{4x}$$
$$= \lim_{x \to \infty} \frac{(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{x}$$
$$= \lim_{x \to \infty} \frac{\frac{1}{\sqrt{x^2}}(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{\frac{1}{\sqrt{x^2}}(x)}$$
$$= \lim_{x \to \infty} \frac{\sqrt{\frac{x^2 + x}{x^2}} + \sqrt{\frac{x^2 - 3x}{x^2}}}{\frac{x}{x}}$$
$$= \lim_{x \to \infty} \frac{\sqrt{1 + \frac{1}{x}} + \sqrt{1 - \frac{3}{x}}}{1}$$
$$= \sqrt{1 + \frac{1}{\infty}} + \sqrt{1 - \frac{3}{\infty}}$$
$$= \sqrt{1 + 0} + \sqrt{1 - 0}$$
$$= \sqrt{1} + \sqrt{1}$$
$$= 1 + 1$$
$$= 2$$

Is this correct?

docnet
Can someone please tell me how to solve a limit problem like this?

$$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$

This is my attempt to solve the problem:

$$\lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}}$$
$$= \lim_{x \to \infty} \frac{4}{\sqrt{x^2 + x} - \sqrt{x^2 - 3x}} × \frac{\sqrt{x^2 + x} + \sqrt{x^2 - 3x}}{\sqrt{x^2 + x} + \sqrt{x^2 - 3x}}$$
$$= \lim_{x \to \infty} \frac{4(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{(x^2 + x) - (x^2 - 3x)}$$
$$= \lim_{x \to \infty} \frac{4(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{x^2 + x - x^2 + 3x}$$
$$= \lim_{x \to \infty} \frac{4(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{4x}$$
$$= \lim_{x \to \infty} \frac{(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{x}$$
$$= \lim_{x \to \infty} \frac{\frac{1}{\sqrt{x^2}}(\sqrt{x^2 + x} + \sqrt{x^2 - 3x})}{\frac{1}{\sqrt{x^2}}(x)}$$
$$= \lim_{x \to \infty} \frac{\sqrt{\frac{x^2 + x}{x^2}} + \sqrt{\frac{x^2 - 3x}{x^2}}}{\frac{x}{x}}$$
$$= \lim_{x \to \infty} \frac{\sqrt{1 + \frac{1}{x}} + \sqrt{1 - \frac{3}{x}}}{1}$$
$$= \sqrt{1 + \frac{1}{\infty}} + \sqrt{1 - \frac{3}{\infty}}$$
$$= \sqrt{1 + 0} + \sqrt{1 - 0}$$
$$= \sqrt{1} + \sqrt{1}$$
$$= 1 + 1$$
$$= 2$$

Is this correct?
Yes.

I wouldn't have used ##\infty ## like a number, since this is not only wrong, it also supports a careless way of dealing with calculations. You have been so detailed and correct, that it hurts a bit to see ##1/\infty ##. Delete this line and the rest is fine.

suremarc, docnet and Delta2
fresh_42 said:
I wouldn't have used ##\infty ## like a number, since this is not only wrong, it also supports a careless way of dealing with calculations. You have been so detailed and correct, that it hurts a bit to see ##1/\infty ##. Delete this line and the rest is fine.

So what I suppose to do after deleting the line? Please show me the proper math write.

So what I suppose to do after deleting the line? Please show me the proper math write.
As I said: it is perfectly fine without that line, just erase it. The long way would be
$$\lim_{n \to \infty}\dfrac{\sqrt{1+\dfrac{1}{x}}+\sqrt{1-\dfrac{3}{x}} }{1}=\dfrac{\sqrt{1+\lim_{n \to \infty}\dfrac{1}{x}}+\sqrt{1-3\cdot\lim_{n \to \infty}\dfrac{1}{x}} }{1}=\dfrac{\sqrt{1+0}+\sqrt{1-3\cdot 0}}{1}$$
but nobody writes these steps, although it shows what is going on, and that we use the fact that the square root function is continuous.

SammyS and docnet

## 1. What is a limit problem?

A limit problem is a mathematical concept that involves finding the value that a function approaches as the input approaches a certain value. It is used to analyze the behavior of a function and determine its limit at a specific point.

## 2. Why is it important to solve limit problems?

Solving limit problems is important because it helps us understand the behavior of a function and determine its value at a specific point. It is also a fundamental concept in calculus and is used in various applications in physics, engineering, and economics.

## 3. What is the step-by-step guide for solving a limit problem?

The step-by-step guide for solving a limit problem involves the following steps:

• 1. Identify the function and the point at which the limit is to be evaluated.
• 2. Simplify the function if possible.
• 3. Use direct substitution to evaluate the limit, if possible.
• 4. If direct substitution is not possible, use algebraic manipulation or special limit rules to simplify the function.
• 5. If the limit is still indeterminate, use L'Hopital's rule or other advanced techniques to evaluate it.
• 6. Check the left and right-hand limits to ensure that they are equal.
• 7. Write the final answer and provide a brief explanation.

## 4. What are some common types of limit problems?

Some common types of limit problems include:

• 1. Limits at a point: finding the limit of a function at a specific point.
• 2. Limits at infinity: determining the behavior of a function as the input approaches infinity.
• 3. Limits involving trigonometric functions: evaluating limits of functions involving trigonometric functions.
• 4. Limits involving rational functions: solving limits of rational functions.
• 5. Limits involving exponential and logarithmic functions: evaluating limits of functions involving exponential and logarithmic functions.

## 5. How can I check if my solution to a limit problem is correct?

You can check if your solution to a limit problem is correct by using a graphing calculator or an online graphing tool to plot the function and see if the limit matches your solution. You can also check if your solution follows the basic rules of limits, such as the sum, difference, product, and quotient rules. Additionally, you can check if your solution is consistent with the left and right-hand limits at the given point.

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