Solving L'Hospital's Rule: lim(x→2)(x^2+x-6)/(x-2)

  • Thread starter phillyolly
  • Start date
In summary, L'Hospital's Rule is a mathematical theorem used to evaluate limits involving indeterminate forms of 0/0 or ∞/∞. It can only be used in these specific cases and involves taking the derivative of both functions, and evaluating the resulting limit. To apply it, first determine if the limit is in an indeterminate form, take the derivatives, and repeat until the limit can be evaluated. The solution to the given limit using L'Hospital's Rule is 5, which can be found by taking the derivatives of the numerator and denominator, resulting in the limit of (2x+1)/(1) as x approaches 2.
  • #1
phillyolly
157
0

Homework Statement



lim(x→2)(x^2+x-6)/(x-2)

Homework Equations





The Attempt at a Solution



lim(x→2)(x^2+x-6)/(x-2)=lim(x→2)(d/dx(x^2+x-6))/(d/dx(x-2))=lim(x→2)(2x+1)/1=lim(x→2)(2x+1)=5

Is this a right solution? I have never done such before, so I am not sure in my answer.
 
Physics news on Phys.org
  • #2
It's fine if you've checked you have a 0/0 form to begin with. You could also factor x^2+x-6 and cancel the denominator to check.
 
  • #3
yeaa, it is a right way,absolutely.L'Hopital's Rule is for 0/0 and inf/inf.
 
  • #4
Thank you a lot, guys.
 

1. What is L'Hospital's Rule?

L'Hospital's Rule is a mathematical theorem that helps to evaluate limits involving indeterminate forms, such as 0/0 or ∞/∞. It states that for a function f(x) and g(x) that approach 0 or ∞ at a given limit, the limit of their quotient can be found by taking the derivative of both functions and evaluating the resulting limit.

2. How do I know if I can use L'Hospital's Rule?

L'Hospital's Rule can only be used when the given limit is in an indeterminate form, specifically 0/0 or ∞/∞. This can be determined by plugging in the value of the limit and seeing if it results in an undefined expression.

3. Can L'Hospital's Rule be used for all types of limits?

No, L'Hospital's Rule can only be used for limits that are in an indeterminate form of 0/0 or ∞/∞. For other types of limits, different methods must be used to evaluate them.

4. How do I apply L'Hospital's Rule to the given limit?

To apply L'Hospital's Rule to a limit, first determine if it is in an indeterminate form of 0/0 or ∞/∞. If it is, take the derivative of the numerator and denominator separately, and then evaluate the resulting limit. If the new limit is still in an indeterminate form, repeat the process until the limit can be evaluated without using L'Hospital's Rule.

5. What is the solution to the given limit using L'Hospital's Rule?

The solution to the given limit is 5. This can be found by taking the derivatives of the numerator and denominator, resulting in the limit of (2x+1)/(1) as x approaches 2. Plugging in the value of 2 for x gives the final answer of (2(2)+1)/(1) = 5.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
237
  • Calculus and Beyond Homework Help
Replies
2
Views
453
  • Calculus and Beyond Homework Help
Replies
15
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
946
  • Calculus and Beyond Homework Help
Replies
6
Views
776
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
782
  • Calculus and Beyond Homework Help
Replies
7
Views
412
  • Calculus and Beyond Homework Help
Replies
7
Views
606
Back
Top