What is the Limit of a Rational Expression?

In summary, the conversation discusses finding the limit of a function as x approaches infinity. The expert summarizer provides a step-by-step explanation of how to solve the problem and mentions that the easier way is to divide everything by ${x}^{2}$. It is also noted that as x approaches infinity, the value of the function approaches the square root of 3. The conversation ends with a thank you and a question about how to initially know which direction to go in when solving this type of problem.
  • #1
karush
Gold Member
MHB
3,269
5
$$\lim_{{x}\to{\infty}}\frac{\sqrt{3x^2 - 1 }}{x-1}=\sqrt{3}$$

I tried dividing everything by ${x}^{2}$ but not
 
Mathematics news on Phys.org
  • #2
karush said:
$$\lim_{{x}\to{\infty}}\frac{\sqrt{3x^2 - 1 }}{x-1}=\sqrt{3}$$

I tried dividing everything by ${x}^{2}$ but not

$$\lim_{x \to +\infty} \frac{\sqrt{3x^2-1}}{x-1}= \lim_{x \to +\infty} \frac{3x^2-1}{(x-1) \sqrt{3x^2-1}}= \lim_{x \to +\infty} \frac{x^2 \left( 3- \frac{1}{x^2}\right)}{(x-1) \sqrt{x^2 \left( 3-\frac{1}{x^2}\right)}}= \lim_{x \to +\infty} \frac{x^2 \left( 3- \frac{1}{x^2}\right)}{(x-1)|x| \sqrt{ \left( 3-\frac{1}{x^2}\right)}}\\= \lim_{x \to +\infty} \frac{x^2 \left( 3- \frac{1}{x^2}\right)}{(x-1)x \sqrt{ \left( 3-\frac{1}{x^2}\right)}}= \lim_{x \to +\infty} \frac{x^2}{x^2 \left(1- \frac{1}{x} \right)} \cdot \lim_{x \to +\infty} \frac{3-\frac{1}{x^2}}{\sqrt{3-\frac{1}{x^2}}}= 1\cdot \frac{3}{\sqrt{3}}= \sqrt{3}$$
 
  • #3
Wow, thanks, that was a lot of steps.
How would you know initially to go in that direction.
 
  • #4
karush said:
Wow, thanks, that was a lot of steps.

(Smile)

karush said:
How would you know initially to go in that direction.
Usually when you have a square root you multiply by it at the numerator and the denominator.
 
  • #5
Could you move this thread to the correct category?
 
  • #6
karush said:
$$\lim_{{x}\to{\infty}}\frac{\sqrt{3x^2 - 1 }}{x-1}=\sqrt{3}$$

I tried dividing everything by ${x}^{2}$ but not

The easier way (first note that I'm leaving out absolute values because everything is positive when going to infinity anyway...)

$\displaystyle \begin{align*} \frac{\sqrt{3x^2 - 1}}{x - 1} &= \frac{\frac{1}{x}\,\sqrt{3x^2 - 1}}{\frac{1}{x} \left( x - 1 \right) } \\ &= \frac{\frac{1}{\sqrt{x^2}}\,\sqrt{3x^2 - 1}}{1 - \frac{1}{x}} \\ &= \frac{\sqrt{\frac{3x^2 - 1}{x^2}}}{1 - \frac{1}{x}} \\ &= \frac{\sqrt{3 - \frac{1}{x^2}}}{1 - \frac{1}{x}} \end{align*}$

So as $\displaystyle \begin{align*} x \to \infty , \, \frac{1}{x} \to 0 \end{align*}$ and $\displaystyle \begin{align*} \frac{1}{x^2} \to 0 \end{align*}$, giving $\displaystyle \begin{align*} \frac{\sqrt{3 - 0}}{1 - 0} = \sqrt{3} \end{align*}$ as the limit :)
 
  • #7
I actually got to your 3rd step before posted, just didn't see the magic.

I sure learn a lot with MHB
 

Related to What is the Limit of a Rational Expression?

What is the definition of a limit of rational expression?

A limit of rational expression is the value that a rational expression approaches as the variable in the expression approaches a certain value. It is often used to determine the behavior of a rational function near a particular point.

How is a limit of rational expression calculated?

To calculate a limit of rational expression, you must first factor the numerator and denominator of the expression. Then, you can evaluate the expression by plugging in the desired value for the variable. If the resulting expression is undefined, you may need to employ algebraic techniques to simplify the expression before evaluating.

What does it mean if the limit of a rational expression does not exist?

If the limit of a rational expression does not exist, it means that the expression does not approach a single value as the variable approaches a particular value. This could be due to a vertical asymptote, a removable discontinuity, or the expression oscillating between two values as the variable gets closer to the desired value.

Can the limit of a rational expression be a complex number?

Yes, the limit of a rational expression can be a complex number. This can occur when the numerator and denominator of the expression contain complex numbers, or when the behavior of the expression near the desired value results in a complex number as the limit.

How are limits of rational expressions used in real-world applications?

Limits of rational expressions are commonly used in fields such as physics, engineering, and economics to model and analyze real-world situations. They can be used to determine maximum and minimum values, predict trends, and evaluate the behavior of systems as variables change.

Similar threads

I Solving Stat Mech Integral with Wolfram Alpha
    • General Math
Replies
2
Views
1K
B Understanding exponentiation and logarithms together
    • General Math
Replies
12
Views
1K
I Expressing any given point on plane with one unique number
    • General Math
Replies
4
Views
719
B Complete the square, yes, but what does it mean?
    • General Math
Replies
19
Views
2K
B Proof That ##\sqrt{x}## Isn't Rational (Unless ##x## is a Perfect Square)
    • General Math
Replies
7
Views
2K
I Why does the integral of sine of x^2 from - infinity to + infinity diverge?
    • General Math
Replies
4
Views
1K
I The Fundamental Theorem of Arithmetic and Rational Numbers
    • General Math
Replies
6
Views
1K
Calculating Left Hand Limit: ##\sqrt 6##
    • Precalculus Mathematics Homework Help
Replies
17
Views
797
MHB Equation 2: Prove that ##x^2+2x\sqrt x+3x+2\sqrt x+1=0##
    • General Math
Replies
4
Views
884
Finding the limit of a rational expression
    • General Math
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top