If determine $$ \lim _{x \rightarrow 0} f(x)=-1 x+0 \leq f(x) \leq x^{2}-1 x+0 $$ What theorem did you use to arrive at your answer? SQUEEZE THEOREM Note: You can earn partial credit on this problem.

Question

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We are given that

$$
-x \leq f(x) \leq x^2 - x
$$

and we need to determine $$\lim_{x \rightarrow 0} f(x)$$.

**Step 1. Compute the limit of the lower bound.**

For the lower bound, we have

$$
\lim_{x \rightarrow 0} (-x) = 0.
$$

**Step 2. Compute the limit of the upper bound.**

For the upper bound, we have

$$
\lim_{x \rightarrow 0} (x^2 - x) = \lim_{x \rightarrow 0}(-x + x^2) = -0 + 0 = 0.
$$

**Step 3. Apply the Squeeze Theorem.**

Since

$$
\lim_{x\to 0}(-x) = \lim_{x\to 0}(x^2 - x) = 0,
$$

by the Squeeze Theorem we conclude that

$$
\lim_{x \rightarrow 0} f(x) = 0.
$$

**Step 4. Identify the theorem used.**

The theorem used to arrive at this answer is the **Squeeze Theorem**.
The Squeeze Theorem was used to determine that $$\lim_{x \rightarrow 0} f(x) = 0$$.

If
determine
What theorem did you use to arrive at your answer?
SQUEEZE THEOREM
Note: You can earn partial credit on this problem.

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If determine What theorem did you use to arrive at your answer? SQUEEZE THEOREM Note: You can earn partial credit on this problem.