Question

What is the limit as x approaches infinity of the function sin(x+1/x) and how do you solve that?

Answer 1

\[\lim_{x\to\infty}\sin\left(x+\frac{1}{x}\right)~~~\text{or}~~~\lim_{x\to\infty}\sin\frac{x+1}{x}~~~?\]

Answer 2

\[\lim_{x \rightarrow \infty}\sin \left( x +\frac{ 1 }{ x } \right)\]

Answer 3

As \(x\to\infty\), do you understand why \(x\to\infty\) and \(\dfrac{1}{x}\to0\) ?

Answer 4

yes. I think what is sin, because as 1/x approaches infinity is staying at 1 and the denominator is blowing up. x is going on to infinity...right?

Answer 5

I'm not sure I understand what you're saying...

My point is that as \(x\to\infty\), you essentially have \(\sin\left(x+\dfrac{1}{x}\right)=\sin x\).

So you have to find \(\displaystyle\lim_{x\to\infty}\sin x\).

Answer 6

\[\lim_{x \rightarrow x} \sin x =1\]
right?

Answer 7

No, that's not it. As x gets larger and larger, \(\sin x\) will constantly alternate between -1 and 1, without resting at a specific value. What does that tell you about the limit?

Answer 8

the limit does not exist because it's not approaching a specific value

Answer 9

Right!

Answer 10

Thank you!