# Math Help - How to solve this limit?

1. ## How to solve this limit?

Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this.

Evaluate:

lim
x -> infinity x sin (1/x)

So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x)
However thats assuming both function's limits exist.

If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0.

Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way.

2. ## Re: How to solve this limit?

Originally Posted by Kuma
Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this.

Evaluate:

lim
x -> infinity x sin (1/x)

So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x)
However thats assuming both function's limits exist.

If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0.

Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way.
Let y= 1/x. Then the problem becomes $\lim_{y\to 0}\frac{sin(y)}{y}$ which is a well known limit.

3. ## Re: How to solve this limit?

Originally Posted by Kuma
Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this.

Evaluate:

lim
x -> infinity x sin (1/x)

So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x)
However thats assuming both function's limits exist.

If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0.

Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way.
This question has been asked many times in these forums, most recently here: http://www.mathhelpforum.com/math-he...ts-188132.html