Does L'Hopital's Rule Apply to Limits Involving Trigonometric Functions?

In summary, the problem involves finding the limit of a function as x approaches infinity. The first part can be solved using the addition/subtraction law, but the second part requires rationalizing the numerator. This can be done by multiplying both the numerator and denominator by a certain expression. This method is preferred over using L'Hopital's rule.
  • #1
Sleighty
5
0

Homework Statement



lim x -> infinity : sqrt(x^2 + 4x(cos x) ) - x

find the limit (or lack there of)

Homework Equations



look above

The Attempt at a Solution



ok so i used the addition/subtraction law to show that the limit of f(x) = - x as x --> infinity = infinity

now for the other half of the function, i can't seem to find out how to mathematically prove that there is no limit. logically i can tell that there is no limit because COS X has no limit.

can someone explain how i prove this mathematically?
 
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  • #2
You are completely wrong about this problem- you cannot use the "addition/subtraction" law here because you cannot add/subtract "infinity".

Think of this as the fraction
[tex]\frac{\sqrt{x^2+ 4x}- x(cos(x))}{1}[/tex]
and "rationalize the numerator"- multiply numerator and denominator by
[tex]\sqrt{x^2+ 4x(cos(x)}+ x[/tex].
 
Last edited by a moderator:
  • #3
thanks mate :) I am pretty noob at calculus. :(

EDIT: what happened to the cos x in the equation?
 
  • #4
Sorry I accidently dropped it. I have edited my previous posts:
Multiply numerator and denominator by
[tex]\sqrt{x^2+ 4x(cos(x)}+ x[/tex].
 
  • #5
Hi, thanks for correcting that mistake, but should i apply L'Hopitals rule in this case? i can't tell :(
 

Related to Does L'Hopital's Rule Apply to Limits Involving Trigonometric Functions?

1. What is a limit?

A limit is the value that a function approaches as the input approaches a certain value. It represents the behavior of a function as the input gets closer and closer to a specific value.

2. How do we find the limit of a function with a square root?

To find the limit of a function with a square root, we can use algebraic manipulation, such as rationalizing the denominator or factoring, to simplify the function. Then, we can plug in the limiting value to find the limit.

3. Can the limit of a function with a square root be undefined?

Yes, the limit of a function with a square root can be undefined. This can occur when the function has a vertical asymptote, meaning that the function approaches infinity or negative infinity as the input approaches a certain value.

4. How do we handle limits involving square roots at infinity?

To handle limits involving square roots at infinity, we can use the concept of the limit at infinity, which is the value that a function approaches as the input gets larger and larger. We can also use L'Hopital's rule to simplify the function and evaluate the limit.

5. Can we use the squeeze theorem to evaluate limits with square roots?

Yes, we can use the squeeze theorem to evaluate limits with square roots. This theorem states that if a function is squeezed between two other functions that approach the same limit, then the function being squeezed must also approach the same limit.

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