(ii) Using a carefully justified application of the Squeeze Theorem, find $\lim_{x \to 0} \left( x^7 \sin \left( \frac{1}{x} \right) \right)$ (3 points)

**Q2**Evaluate $\lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{x}$ You should show your reasoning carefully, however you may use any of the limit laws without explanation or proof.

**Q3**(i) Let $f$ be a real-valued function, and let $a$ and $L$ be real numbers. What does it mean to say that $\lim_{x \to a} f(x) = L$? (2 points)

Prove carefully, using the definition of $\lim_{x \to a} f(x) = L$, that
$\lim_{x \to 3}(x^2 + 2) = 11$

**Q4**(i) State carefully the Intermediate Value Theorem (2 points)

(ii) Prove that there is a root of the equation $2x^3 = 3^x$ in the interval $(1, 2)$. (3 points)

**Q5**The figure below shows the graph of $y = f(x)$ when $f(x) = \begin{cases} \frac{5}{x + 1} & \text{if} \; x < -\pi \\ \tan (x) & \text{if} \; -\pi \le x \lt \; \text{but} \; x \ne -frac{\pi}{2} \\ 0 & \text{if} \; x = -\frac{\pi}{2} \\ x^2 & \text{if} \; 0 \le x \lt 1 \\ 3 & \text{if} \; 1 \le x \lt 4 \\ 1 & \text{if} \; x = 4 \\ \frac{1}{(x - 5)^2} + 2 & \text{if} \; 4 < x < 7 \; \text{but} \; x \ne 5 \\ -2 & \text{if} \; x = 5 \\ \ln (x) & \text{if} \; x \ge 7 \end{cases}$

For each of the following statements, indicate if it is true or false.

(i) $\lim_{x \to 4} f(x) = 3$

(ii) $\lim_{x \to 5^{+}} f(x) = \infty$

$\lim_{x \to 5} f(x) = \infty$

$\lim_{x \to -\pi} f(x)$ exists

$\lim_{x \to -\frac{\pi}{2}^2} f(x) = \infty$

$\lim_{x \to \infty} f(x) = \infty$

The graph $y = f(x)$ has a horizontal asymptote at $y = 0$.

The graph $y = f(x)$ has two horizontal asymptotes.

The graph $y = f(x)$ has two vertical asymptotes.

$f(x)$ is continuous at $x = 0$.

$f(x)$ is continuous at $x = 1$

$f(x)$ is continuous on the interval $[1, 2]$.

**Q6**Using the limit definition of derivative, show that if $f(x) = x^2$, then $f'(x) = 2x$. (2 points)

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