Understanding Trig Limits and Differentiation Rules

[esploded]

Homework Statement

1. sin(1/x) as x->infinity
2. tan(7x)/sin(5x) as x->0
3. General question--when finding the derivative (through implicit diff.), and I have something like y=(5x^2)y, what parts am I supposed to use the chain rule, and when am I supposed to use the product rule?

The Attempt at a Solution

1. I literally have no idea how to do this. Possibly multiplying, but sin/sinx doesn't seem to get me anywhere.
2. sin7x/cos7x times 1/sin5x. Not quite sure what to do afterwards.
3. Is this correct? -- 10x^2(y) times y' ?

 

[kru_]

1: Consider what the limit of 1/x is, as x tends toward infinity. Then use that to evaluate sin(1/x) as x tends toward infinity.

2: Consider using L'Hopital's Rule.

3: I am not certain that I fully understand what you're asking, so I will refrain from comment.

 

[esploded]

1: Consider what the limit of 1/x is, as x tends toward infinity. Then use that to evaluate sin(1/x) as x tends toward infinity.

2: Consider using L'Hopital's Rule.

3: I am not certain that I fully understand what you're asking, so I will refrain from comment.

I'm honestly unsure how to explain this as well.
I just need help on how to do number 3.

 

[SammyS]

Look at 1/x when x = BIG number. Try 1000, then 1000000, then 10¹⁰⁰, etc.