Can you help me evaluate this limit?

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  • #1
HF08
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[SOLVED] Can you help me evaluate this limit?

lim k -[tex]\sqrt{k^{2}+k}[/tex] as k[tex]\rightarrow[/tex][tex]\infty[/tex]

I have evaluated the limit using computer technology and I know it should be
-1/2. I have tried using something like the squeeze theorem but failed.

My other attempt is to mutiply the limit by the expression
lim k +[tex]\sqrt{k^{2}+k}[/tex] / lim k +[tex]\sqrt{k^{2}+k}[/tex]

Please help me. I know this is a simple problem, but I got stuck. Thank you
for your time and I hope you reply.

HF08
 
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  • #2
[tex]\lim_{k\rightarrow \infty}(k-\sqrt{k^2+k})[/tex]

[tex]\lim_{k\rightarrow \infty}\left(k-\sqrt{k^2+k} \cdot \frac{k+\sqrt{k^2+k}}{k+\sqrt{k^2+k}}\right)[/tex]

[tex]\lim_{k\rightarrow \infty}\left(\frac{-k}{k+\sqrt{k^2+k}}\right)[/tex]
 
Last edited:
  • #3
Hf08

[tex]\lim_{k\rightarrow \infty}(k-\sqrt{k^2+k})[/tex]

This should be right...
 
  • #4
Almost there...

rocophysics said:
[tex]\lim_{k\rightarrow \infty}(k-\sqrt{k^2+k})[/tex]

[tex]\lim_{k\rightarrow \infty}\left(k-\sqrt{k^2+k} \cdot \frac{k+\sqrt{k^2+k}}{k+\sqrt{k^2+k}}\right)[/tex]

[tex]\lim_{k\rightarrow \infty}\left(\frac{-k}{k+\sqrt{k^2+k}}\right)[/tex]

Right. Now, this gives us [tex]\lim_{k\rightarrow \infty}\left(\frac{-1}{1+\frac{\sqrt{k^2+k}}{k}}\right)[/tex]

So, the next question is how to show that [tex]\lim_{k\rightarrow \infty}\left(\frac{\sqrt{k^2+k}}{k}}\right)[/tex] = 1 . I tried l'hopital, but I wonder if there is another way?
I think is is fairly obvious this is basically k/k for large k, but how to prove that with rigor?

Thanks,
HF08
 
  • #5
[tex]\lim_{k\rightarrow \infty}\left(\frac{-1}{1+\sqrt{\frac{k^2+k}{k^2}}}\right)[/tex]

[tex]\lim_{k\rightarrow \infty}\left(\frac{-1}{1+\sqrt{1+\frac 1 k}}\right)[/tex]
 
  • #6
Thanks

rocophysics said:
[tex]\lim_{k\rightarrow \infty}\left(\frac{-1}{1+\sqrt{\frac{k^2+k}{k^2}}}\right)[/tex]

[tex]\lim_{k\rightarrow \infty}\left(\frac{-1}{1+\sqrt{1+\frac 1 k}}\right)[/tex]

(Slaps forehead). Thanks Rocophyics. Can I modify this thread to show solved? I am still new to this forum.
 
  • #7
It's ok! Yeah, go to Thread Tools (least I think so, I never do it myself :p)
 

Related to Can you help me evaluate this limit?

1. What is a limit in mathematics?

A limit in mathematics is the value that a function or sequence approaches as the input or index approaches a specific value. It can be thought of as the value that the function "gets close to" as the input gets closer and closer to a particular value.

2. How do you evaluate a limit?

To evaluate a limit, you need to plug in the given value into the function and see what output you get. If the output is undefined or doesn't exist, you may need to use other methods such as algebraic manipulation, graphing, or L'Hôpital's rule to evaluate the limit.

3. What are the common types of limits?

The common types of limits include finite limits, infinite limits, one-sided limits, and limits at infinity. Finite limits are when the output of the function approaches a specific value as the input gets close to a particular value. Infinite limits are when the output of the function approaches positive or negative infinity as the input gets close to a particular value. One-sided limits are when the input approaches a specific value from only one side. Limits at infinity are when the input approaches positive or negative infinity.

4. What are the different notations for limits?

The different notations for limits include using the limit symbol, "lim", with the value the input is approaching written below it. For example, lim x→a. Another notation is using an arrow symbol, →, to show the direction the input is approaching a particular value. For example, x→a as x approaches a. Lastly, there is the epsilon-delta notation, which uses the Greek letters epsilon (ε) and delta (δ) to show the closeness of the input and output values in a limit.

5. Why are limits important in mathematics?

Limits are important in mathematics because they allow us to analyze the behavior of functions and sequences. They help us determine the continuity and differentiability of a function, as well as understand the behavior of a function at different points. Limits also play a crucial role in calculus, where they are used to find derivatives and integrals.

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