Here is a summary of the answers marked on your paper:

1. Answer: D) $\frac{5}{4}$
2. Answer: B) Does not exist
3. Answer: C) $0$
4. Answer: B) $x = -2, x = 0$
5. Answer: B) $-4$

Observations and Steps for Each Question:

1. Limits involving trigonometric functions: The user marked $\frac{5}{4}$, possibly requiring validation.

2. $\lim_{x \to 0} x \cos\left(\frac{1}{x}\right)$:
   Using the Squeeze Theorem: $-x \leq x\cos\left(\frac{1}{x}\right) \leq x$, and as $x \to 0$, both bounds approach $0$. Therefore, the limit exists and equals $0$.
   Correct Answer: $C) 0$, not $B$ as marked.

3. $x^3 \leq f(x) \leq x$ for $x \in [-1,1]$:
   When $x \to 0$, both $x^3 \to 0$ and $x \to 0$, so $f(x) \to 0$.
   Correct Answer: $C) 0$ (user answer is correct).

4. Discontinuities:
   User identified $x = -2$ and $x = 0$ based on visual inspection, which is accurate.
   Correct Answer: $B) x = -2, x = 0$.

5. Remove discontinuity by adjusting $f(1)$:
   To make $f(x)$ continuous, equating limits as $x \to 1$:
   $\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x).$ Calculation shows $f(1) = -4$.
   Correct Answer: $B) -4$ (user answer is correct).

Relative Questions:

1. How does the Squeeze Theorem work in evaluating limits?
2. What conditions must be satisfied for a function to be continuous?
3. How do you identify points of discontinuity from a graph?
4. What are the key steps to solve piecewise continuity problems?
5. How are trigonometric limits evaluated as $x \to 0$?

Tip: Always verify the Squeeze Theorem limits by analyzing both upper and lower bounds carefully.

Let me know if you need detailed solutions for any of these problems!