Lim sqrt(x)/sqrt(10x+1) ?? lim sqrt(x)/sqrt(10x+1) ??

  • Thread starter IntegrateMe
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In summary, the limit of sqrt(x)/sqrt(10x+1) as x tends to infinity is equal to sqrt(1/10). To solve this, we can simplify the expression by factoring out x from both terms in the denominator. Then, we can set x to infinity and evaluate the limit, which is equal to 1. The radical sign was removed during simplification and we cannot use infinity/infinity or infinity - infinity as the answer.
  • #1
IntegrateMe
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lim sqrt(x)/sqrt(10x+1) ??

Limit as x tends to infinity of sqrt(x)/sqrt(10x+1) = sqrt(1/10), but how?

I'm trying to understand this, so, it's the sqrt[x/(10x+1)], but how do i simplify that to make it a limit i can set to infinity. I know the problem isn't my calculus skills, but my algebra skills!
 
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  • #2


Factor x out of both terms in the denominator.
[tex]\frac{x}{10x + 1}~=~\frac{x}{x(10 + 1/x)}[/tex]

Now, what's the limit of the latter expression as x gets large?
 
  • #3


If we set x to infinity we get infinity/infinity(10) which is just equal to infinity/infinity?? The limit as x gets very large is 1 though.

Why did you take off the radical sign, btw, is that part of simplification?
 
  • #4


nevermind, i got it. thanks a lot.
 
  • #5


IntegrateMe said:
If we set x to infinity we get infinity/infinity(10) which is just equal to infinity/infinity?? The limit as x gets very large is 1 though.

Why did you take off the radical sign, btw, is that part of simplification?

infinity/infinity or infinity - infinity is NEVER the answer.

I was working inside the radical...
 

1. What is the limit of the function sqrt(x)/sqrt(10x+1) as x approaches infinity?

As x approaches infinity, the function approaches 0. This can be seen by dividing both the numerator and denominator by x, resulting in sqrt(1)/sqrt(10+1/x), which simplifies to 1/sqrt(10). As x gets larger and larger, 1/x approaches 0, making the denominator approach sqrt(10), which results in a limit of 0.

2. Is this function continuous at x=0?

Yes, this function is continuous at x=0. This can be seen by evaluating the limit of the function as x approaches 0 from the left and right. Both limits will approach 0, resulting in a limit at x=0 that equals 0. Therefore, the function is continuous at x=0.

3. Does this function have any vertical asymptotes?

No, this function does not have any vertical asymptotes. As x approaches infinity, the function approaches 0, and as x approaches negative infinity, the function also approaches 0. Therefore, there are no values of x that would result in a vertical asymptote.

4. Can this function be simplified?

Yes, this function can be simplified by multiplying the numerator and denominator by sqrt(10x+1). This results in (sqrt(x)*(sqrt(10x+1))/sqrt(10x+1)^2, which simplifies to sqrt(x)/(10x+1).

5. What is the domain of this function?

The domain of this function is all real numbers except for x=-1/10. This is because when x=-1/10, the denominator becomes 0, resulting in a undefined value. Therefore, the domain is (-∞,-1/10)U(-1/10,∞).

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