Evaluating the limit of a function

In summary, the student is trying to find an equation for the limit as x goes to zero, but is lost and needs help. They tried manipulating the equation to a point where it could be simplified, but then found that Taylor series could be used to approximate the function. They then confirmed the answer by doing the math from there.
  • #1
moouers
80
0

Homework Statement


I'm not sure how to type "lim x-> 0 out using this forum's math symbols...

lim x→0 [[itex]\frac{1}{x\sqrt{1+x}}[/itex]-[itex]\frac{1}{x}[/itex]]


Homework Equations





The Attempt at a Solution



Honestly, I'm just not seeing how to manipulate the equation to get it to a point where I can find the solution. This isn't graded and isn't even for any class I'm in...I'm just curious. It's been 10 years since I've done anything like this, so I'm trying to get up to speed before I take Calculus again in the fall.
 
Physics news on Phys.org
  • #2
Since x is going to zero, I'd try a taylor series about x=0 ;)
 
  • #3
Haha, well, I'd love to. But since this is in one of the first few chapters in the book, long before Taylor series is introduced, there must be another way and I'd like do it that way first.
 
  • #4
oh.. :p

In that case, try getting everything into one term, with a common denominator and see what happens
 
  • #5
I tried that and was just as lost. Maybe I did it wrong, but I got:

lim x→0 [[itex]\frac{1-\sqrt{1+x}}{x\sqrt{1+x}}[/itex]]

If I did that correctly, I still don't know where to go from there.
 
  • #6
also try multiplying by 1 in a nice way :3
 
Last edited:
  • #7
Would you mind showing my the first step to take, so I can go on from there? I would appreciate it.
 
  • #8
moouers said:
I tried that and was just as lost. Maybe I did it wrong, but I got:

lim x→0 [[itex]\frac{1-\sqrt{1+x}}{x\sqrt{1+x}}[/itex]]

If I did that correctly, I still don't know where to go from there.

Now try multiplying numerator and denominator by [itex]1+\sqrt{1+x}[/itex]. Expand out the numerator.
 
  • #9
No, the expansion was done correctly.

Try rationalizing the numerator by multiplying by ##\frac{1 + \sqrt{1 + x}}{1 + \sqrt{1 + x}}##

Also, Taylor series are used for approximating functions, not evaluating limits.

Edit: ninjas on this board!
 
  • #10
Wow, so that was the trick. I carried out the math from there and caclulator/book confirms the answer. Thank you all so much. That was far simpler than it seemed.
 

1. What is a limit of a function?

A limit of a function is a fundamental concept in calculus that describes the behavior of a function as its input value approaches a specific value. It represents the value that a function approaches, but may not necessarily reach, as the input value gets closer and closer to the specified value.

2. Why is evaluating the limit of a function important?

Evaluating the limit of a function is important because it allows us to understand the behavior of the function and make predictions about its values. It also helps us determine if a function is continuous at a particular point, which is essential in many real-life applications such as engineering, physics, and economics.

3. How do you evaluate the limit of a function algebraically?

To evaluate the limit of a function algebraically, we can use various techniques such as direct substitution, factoring, rationalization, and the use of limit laws. These algebraic methods involve manipulating the function to simplify its expression and then plugging in the specified value of the input variable to find the limit.

4. Can the limit of a function have a different value from the function's actual value?

Yes, the limit of a function can have a different value from the function's actual value, especially if the function is discontinuous at the specified point. In this case, the limit represents the behavior of the function as it approaches the specified value, while the actual value of the function may not exist or may be different from the limit.

5. Are there any limitations to evaluating the limit of a function?

Yes, there are some limitations to evaluating the limit of a function. One limitation is that a function must be defined and continuous at the specified point for its limit to exist. Another limitation is that some functions may have a limit that cannot be evaluated using algebraic methods, and numerical or graphical methods must be used instead.

Similar threads

Interval of convergence of power series from ratio test
    • Calculus and Beyond Homework Help
Replies
2
Views
187
Spivak, Ch 5 Limits, Problems 10c: Proving limit relationship
    • Calculus and Beyond Homework Help
Replies
5
Views
876
Find limit of multi variable function
    • Calculus and Beyond Homework Help
Replies
10
Views
827
Real analysis: prove the limit exists
    • Calculus and Beyond Homework Help
Replies
13
Views
2K
Use Graph to Determine Limit: Calculating Limits with Piecewise Functions
    • Calculus and Beyond Homework Help
Replies
3
Views
731
Limit problem not making sense
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
How to show that these sequences are summable?
    • Calculus and Beyond Homework Help
Replies
1
Views
257
Prove that this series converges uniformly on R
    • Calculus and Beyond Homework Help
Replies
4
Views
308
Limit of 0^0: Evaluating x^sinx
    • Calculus and Beyond Homework Help
Replies
19
Views
2K
Evaluating the limit of a multivariable function with paths?
    • Calculus and Beyond Homework Help
Replies
8
Views
1K

Hot Threads

Recent Insights

Back
Top