Solving Limit Without L'Hospital's Rule

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In summary, the conversation was about solving a limit problem using L'hospital's rule, but the person wanted to know another method. It was discovered that the numerator and denominator had a common factor of (x-2), which could be cancelled out to solve the problem. Polynomial division or synthetic division could also be used to solve for the limit.
  • #1
chislam
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I was working out a limit problem and even though I solved it using L'hospital's rule, I do not know how to solve it another way which was probably the intended way. Here's the limit:

[tex]\lim_{x\rightarrow2} \frac{x^4 - x^3 + x^2 - 4x - 4}{2x^2 - 5x + 2}[/tex]

So I used L'hospital's rule and got 20/3 but I want to know another method.

Thanks
 
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  • #2
The numerator and denominator have a common factor of (x-2) that is causing it to have a 0/0 form. If you cancel out that common factor, you can just put x=2.
 
  • #3
Yeah I figured that was the case, but I wasn't able to factor it out in the numerator.
 
  • #4
Just do polynomial division of (x-2) into the numerator. You know you'll get zero remainder, right?
 
  • #5
Doh! Lol, I completely forgot about synthetic division.
 

FAQ: Solving Limit Without L'Hospital's Rule

What is L'Hospital's Rule and why is it used to solve limits?

L'Hospital's Rule is a mathematical technique used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It is based on the concept of derivatives and allows us to convert a difficult limit problem into a simpler one.

Can limits be solved without using L'Hospital's Rule?

Yes, there are other methods for solving limits, such as direct substitution, factoring, and trigonometric identities. L'Hospital's Rule is just one of many tools available to us.

What are the steps for solving a limit without using L'Hospital's Rule?

The first step is to try to simplify the expression as much as possible by using algebraic manipulation or trigonometric identities. Then, we can try to use substitution or factoring to get rid of any indeterminate forms. If those methods do not work, we can try to use special limit theorems, such as the Squeeze Theorem or the Sandwich Theorem. If all else fails, we can use numerical methods, such as a graphing calculator or a table of values, to approximate the limit.

Are there any limitations to using L'Hospital's Rule?

Yes, L'Hospital's Rule can only be applied to limits of indeterminate forms. If the limit is not an indeterminate form, then L'Hospital's Rule cannot be used. Additionally, the rule assumes that the limit exists and is finite, so it cannot be used to evaluate limits that do not exist or are infinite.

When is it appropriate to use L'Hospital's Rule to solve a limit?

L'Hospital's Rule is most useful when evaluating limits involving rational functions, trigonometric functions, and exponential functions. It is also helpful when the limit involves a product or quotient of functions. However, it is important to note that L'Hospital's Rule should only be used as a last resort, after trying other methods of solving the limit.

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