Question

Use the Squeeze Theorem ​

Answer 1

See Below.

Explanation:

We want to use the Squeeze Theorem to show that:

`\displaystyle \lim_(x \to 0)\left(x^2\sin\left((2)/(x)\right)\right)=0`

Recall that according to the Squeeze Theorem, if:

`\displaystyle g(x)\leq f(x) \leq h(x)`

And:

`\displaystyle \lim_(x\to c)g(x) =\lim_(x\to c)h(x) = L`

Then:

`\displaystyle \lim_(x\to c)f(x)=L`

Recall that the value of sine is always ≥ -1 and ≤ 1. Hence:

`\displaystyle -1 \leq \sin\left((2)/(x)\right) \leq 1`

We can multiply both sides by x². Since this value is always positive, we do not need to change the signs. Hence:

`\displaystyle -x^2\leq x^2\sin\left((2)/(x)\right)\leq x^2`

Let g = -x², h = x², and f = x²sin(2 / x). We can see that:

`\displaystyle \lim_(x \to 0)g(x) = \lim_( x \to 0)h(x) = 0`

And since g(x) ≤ f(x) ≤ h(x), we can conclude using the Squeeze Theorem that:

`\displaystyle \lim_(x \to 0)f(x) = \lim_(x \to 0)x^2\sin\left((2)/(x)\right)=0`