5 5.8 Use the graphs of $$ f $$ and $$ g $$ (not calculations) to solve $$ f(x) \cdot g(x) \geq 0 $$ QUESTION 6 6.1 Evaluate the following limits: (a) $$ \lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{3}+6\right) $$ (b) $$ \lim _{u \rightarrow-2} \sqrt{u^{2}+3 u+6} $$ (c) Let $$ f(x)=\frac{2 x+12}{|x+6|} $$ Determine: (i) $$ \lim _{x \rightarrow 6^{+}} f(x) $$ (ii) $$ \lim _{x \rightarrow 6^{-}} f(x) $$ (iii) Use (i) and (ii) to deduce $$ \lim f(x) $$ at $$ x=6 $$. 6.2 Let $$ f(x)=\left\{\begin{array}{ccc} \cos x & \text { if } & x0 \end{array}\right. $$

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5 5.8 Use the graphs of $$ f $$ and $$ g $$ (not calculations) to solve $$ f(x) \cdot g(x) \geq 0 $$ QUESTION 6 6.1 Evaluate the following limits: (a) $$ \lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{3}+6\right) $$ (b) $$ \lim _{u \rightarrow-2} \sqrt{u^{2}+3 u+6} $$ (c) Let $$ f(x)=\frac{2 x+12}{|x+6|} $$ Determine: (i) $$ \lim _{x \rightarrow 6^{+}} f(x) $$ (ii) $$ \lim _{x \rightarrow 6^{-}} f(x) $$ (iii) Use (i) and (ii) to deduce $$ \lim f(x) $$ at $$ x=6 $$. 6.2 Let $$ f(x)=\left\{\begin{array}{ccc} \cos x & \text { if } & x<0 \ 0 & \text { if } & x=0 \ 1-x^{2} & \text { if } & x>0 \end{array}\right. $$

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### Question 5: Solve $$ f(x) \cdot g(x) \geq 0 $$

To solve the inequality $$ f(x) \cdot g(x) \geq 0 $$ using the graphs of $$ f $$ and $$ g $$, we need to identify the intervals where the product of the two functions is non-negative.

1. **Identify the zeros of $$ f(x) $$ and $$ g(x) $$**: 
   - Find the points where $$ f(x) = 0 $$ and $$ g(x) = 0 $$. These points will divide the x-axis into intervals.

2. **Determine the sign of $$ f(x) $$ and $$ g(x) $$** in each interval:
   - For each interval created by the zeros, check the sign of $$ f(x) $$ and $$ g(x) $$. 
   - The product $$ f(x) \cdot g(x) $$ will be non-negative if both functions are either positive or both are negative.

3. **Combine the intervals**:
   - List the intervals where $$ f(x) \cdot g(x) \geq 0 $$.

### Question 6: Evaluate the limits

#### 6.1 Evaluate the following limits:

(a) $$ \lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{3}+6\right) $$

To evaluate this limit, we can substitute $$ x = 2 $$:

$$
\lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{3}+6\right) = (2^2 + 2)(3 \cdot 2^3 + 6)
$$

Calculating this step-by-step:

1. Calculate $$ 2^2 + 2 = 4 + 2 = 6 $$.
2. Calculate $$ 3 \cdot 2^3 + 6 = 3 \cdot 8 + 6 = 24 + 6 = 30 $$.
3. Multiply the results: $$ 6 \cdot 30 = 180 $$.

Thus,

$$
\lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{3}+6\right) = 180.
$$

(b) $$ \lim _{u \rightarrow-2} \sqrt{u^{2}+3 u+6} $$

Substituting $$ u = -2 $$:

$$
\lim _{u \rightarrow-2} \sqrt{u^{2}+3 u+6} = \sqrt{(-2)^{2} + 3(-2) + 6}
$$

Calculating this step-by-step:

1. Calculate $$ (-2)^{2} = 4 $$.
2. Calculate $$ 3(-2) = -6 $$.
3. Combine: $$ 4 - 6 + 6 = 4 $$.
4. Take the square root: $$ \sqrt{4} = 2 $$.

Thus,

$$
\lim _{u \rightarrow-2} \sqrt{u^{2}+3 u+6} = 2.
$$

(c) Let $$ f(x)=\frac{2 x+12}{|x+6|} $$

(i) $$ \lim _{x \rightarrow 6^{+}} f(x) $$

For $$ x \rightarrow 6^{+} $$, $$ |x+6| = x + 6 $$:

$$
\lim _{x \rightarrow 6^{+}} f(x) = \lim _{x \rightarrow 6^{+}} \frac{2x + 12}{x + 6} = \frac{2(6) + 12}{6 + 6} = \frac{12 + 12}{12} = \frac{24}{12} = 2.
$$

(ii) $$ \lim _{x \rightarrow 6^{-}} f(x) $$

For $$ x \rightarrow 6^{-} $$, $$ |x+6| = x + 6 $$:

$$
\lim _{x \rightarrow 6^{-}} f(x) = \lim _{x \rightarrow 6^{-}} \frac{2x + 12}{x + 6} = \frac{2(6) + 12}{6 + 6} = \frac{12 + 12}{12} = \frac{24}{12} = 2.
$$

(iii) Since both one-sided limits are equal:

$$
\lim_{x \rightarrow 6} f(x) = 2.
$$

#### 6.2 Evaluate the piecewise function

For the function

$$
f(x)=\left\{\begin{array}{ccc}
\cos x & \text { if } & x<0 \
0 & \text { if } & x=0 \
1-x^{2} & \text { if } & x>0
\end{array}\right.
$$

To find the limit as $$ x $$ approaches 0:

1. From the left ($$ x \rightarrow 0^{-} $$): $$ f(x) = \cos x \rightarrow \cos(0) = 1 $$.
2. From the right ($$ x \rightarrow 0^{+} $$): $$ f(x) = 1 - x^{2} \rightarrow 1 - 0^{2} = 1 $$.

Since both limits are equal:

$$
\lim_{x \rightarrow 0} f(x) = 1.
$$

### Summary of Results
- $$ f(x) \cdot g(x) \geq 0 $$: Analyze using graphs.
- $$ \lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{3}+6\right) = 180 $$
- $$ \lim _{u \rightarrow-2} \sqrt{u^{2}+3 u+6} = 2 $$
- $$ \lim _{x \rightarrow 6^{+}} f(x) = 2 $$
- $$ \lim _{x \rightarrow 6^{-}} f(x) = 2 $$
- $$ \lim_{x \rightarrow 6} f(x) = 2 $$
- $$ \lim_{x \rightarrow 0} f(x) = 1 $$
- Solve $$ f(x) \cdot g(x) \geq 0 $$ by analyzing the graphs of $$ f $$ and $$ g $$ to find intervals where both functions are positive or both are negative. - $$ \lim _{x \rightarrow 2}\left(x^{2}+x\right)\left(3 x^{3}+6\right) = 180 $$ - $$ \lim _{u \rightarrow-2} \sqrt{u^{2}+3 u+6} = 2 $$ - $$ \lim _{x \rightarrow 6^{+}} f(x) = 2 $$ and $$ \lim _{x \rightarrow 6^{-}} f(x) = 2 $$, so $$ \lim_{x \rightarrow 6} f(x) = 2 $$ - $$ \lim_{x \rightarrow 0} f(x) = 1 $$

5
5.8 Use the graphs of and (not calculations) to solve

QUESTION 6
6.1 Evaluate the following limits:
(a)
(b)
© Let

Determine:
(i)
(ii)
(iii) Use (i) and (ii) to deduce at .
6.2 Let

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5 5.8 Use the graphs of and (not calculations) to solve QUESTION 6 6.1 Evaluate the following limits: (a) (b) © Let Determine: (i) (ii) (iii) Use (i) and (ii) to deduce at . 6.2 Let