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 $$ f(x)=\left\{\begin{array}{cll} \cos x & \text { if } & x0 \end{array}\right. $$

Question

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

UpStudy AI · Accepted Answer

Sure, let's evaluate each part of **Question 6.1** step by step.

---

### **6.1 Evaluate the following limits:**

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

**Solution:**

The function $$ \left(x^{2}+x\right)\left(3 x^{3}+6\right) $$ is a polynomial, which is continuous everywhere. Therefore, we can directly substitute $$ x = 2 $$ into the function to find the limit.

1. **Calculate $$ x^2 + x $$ at $$ x = 2 $$:**
   $$
   2^2 + 2 = 4 + 2 = 6
   $$

2. **Calculate $$ 3x^3 + 6 $$ at $$ x = 2 $$:**
   $$
   3(2)^3 + 6 = 3(8) + 6 = 24 + 6 = 30
   $$

3. **Multiply the two results:**
   $$
   6 \times 30 = 180
   $$

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

---

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

**Solution:**

First, evaluate the expression inside the square root at $$ u = -2 $$:

1. **Calculate $$ u^2 + 3u + 6 $$ at $$ u = -2 $$:**
   $$
   (-2)^2 + 3(-2) + 6 = 4 - 6 + 6 = 4
   $$

2. **Take the square root of the result:**
   $$
   \sqrt{4} = 2
   $$

Since the function inside the limit is continuous at $$ u = -2 $$, the limit is simply the value of the function at that point.

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

---

#### **(c) Let $$ f(x)=\frac{2 x+12}{|x+6|} $$. Determine:**

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

**Solution:**

When $$ x $$ approaches $$ 6 $$ from the right ($$ x \rightarrow 6^{+} $$):

1. **Evaluate the denominator $$ |x + 6| $$:**
   $$
   x + 6 \rightarrow 6 + 6 = 12 \quad (\text{since } x > 6)
   $$
   $$
   |x + 6| = 12
   $$

2. **Evaluate the numerator $$ 2x + 12 $$ at $$ x = 6 $$:**
   $$
   2(6) + 12 = 12 + 12 = 24
   $$

3. **Compute the limit:**
   $$
   \frac{24}{12} = 2
   $$

**Answer:**  
$$
\lim _{x \rightarrow 6^{+}} f(x) = 2
$$

---

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

**Solution:**

When $$ x $$ approaches $$ 6 $$ from the left ($$ x \rightarrow 6^{-} $$):

1. **Evaluate the denominator $$ |x + 6| $$:**
   $$
   x + 6 \rightarrow 6 + 6 = 12 \quad (\text{since } x < 6, \text{ but } x + 6 > 0)
   $$
   $$
   |x + 6| = 12
   $$

2. **Evaluate the numerator $$ 2x + 12 $$ at $$ x = 6 $$:**
   $$
   2(6) + 12 = 12 + 12 = 24
   $$

3. **Compute the limit:**
   $$
   \frac{24}{12} = 2
   $$

**Answer:**  
$$
\lim _{x \rightarrow 6^{-}} f(x) = 2
$$

---

##### **(iii) Use (i) and (ii) to deduce $$ \lim_{x \rightarrow 6} f(x) $$ at $$ x=6 $$.**

**Solution:**

Since both the right-hand limit ($$ 2 $$) and the left-hand limit ($$ 2 $$) are equal, the two-sided limit exists and is equal to this common value.

**Answer:**  
$$
\lim_{x \rightarrow 6} f(x) = 2
$$

---

### **Summary of Answers for Question 6.1:**

- **(a)** The limit is **180**.
- **(b)** The limit is **2**.
- **(c)**
  - **(i)** $$ \lim _{x \rightarrow 6^{+}} f(x) = 2 $$
  - **(ii)** $$ \lim _{x \rightarrow 6^{-}} f(x) = 2 $$
  - **(iii)** $$ \lim_{x \rightarrow 6} f(x) = 2 $$

---

If you have any further questions or need assistance with **Question 6.2**, feel free to ask!
**Question 6.1 Solutions:** - **(a)** The limit is **180**. - **(b)** The limit is **2**. - **(c)** - **(i)** As $$ x $$ approaches 6 from the right, $$ f(x) $$ approaches **2**. - **(ii)** As $$ x $$ approaches 6 from the left, $$ f(x) $$ approaches **2**. - **(iii)** Therefore, the two-sided limit at $$ x = 6 $$ is **2**. **Question 6.2:** Not answered here.

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

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