To solve this problem, we need to determine the continuity of each of the three given piecewise functions across their respective domains. Let's examine each function.

(i) ( f(x) = \begin{cases}

x + 2 & \text{if } x \leq 1 \ 3 + 5 \ln x & \text{if } x > 1 \end{cases} )

• For $x \leq 1$: $f(x) = x + 2$ is a linear function, which is continuous everywhere.
• For $x > 1$: $f(x) = 3 + 5 \ln x$ is continuous wherever $\ln x$ is defined, i.e., for $x > 0$. This ensures continuity for $x > 1$.

Now check the continuity at $x = 1$:

$\lim_{x \to 1^-} f(x) = 1 + 2 = 3$ $\lim_{x \to 1^+} f(x) = 3 + 5 \ln 1 = 3 + 0 = 3$

Since both left and right limits are equal and the function is defined at $x = 1$, $f(x)$ is continuous at $x = 1$.

Thus, function (i) is continuous.

(ii) ( f(x) = \begin{cases}

7e^x - 4 & \text{if } x < 0 \ 3 \cos x & \text{if } x \geq 0 \end{cases} )

• For $x < 0$: $f(x) = 7e^x - 4$ is an exponential function, which is continuous everywhere.
• For $x \geq 0$: $f(x) = 3 \cos x$ is also continuous everywhere.

Now check the continuity at $x = 0$:

$\lim_{x \to 0^-} f(x) = 7e^0 - 4 = 7 - 4 = 3$ $\lim_{x \to 0^+} f(x) = 3 \cos(0) = 3 \times 1 = 3$

Since both left and right limits are equal and the function is defined at $x = 0$, $f(x)$ is continuous at $x = 0$.

Thus, function (ii) is continuous.

(iii) ( f(x) = \begin{cases}

\frac{x^2 - 49}{x - 7} & \text{if } x \neq 7 \ 12 & \text{if } x = 7 \end{cases} )

• For $x \neq 7$: We can simplify the given expression as follows: $f(x) = \frac{x^2 - 49}{x - 7} = \frac{(x - 7)(x + 7)}{x - 7} = x + 7 \quad \text{for } x \neq 7$ This is a linear function, which is continuous everywhere except possibly at $x = 7$.

Now check the continuity at $x = 7$:

$\lim_{x \to 7} f(x) = 7 + 7 = 14$ But $f(7) = 12$, so there is a jump discontinuity at $x = 7$.

Thus, function (iii) is not continuous.

Conclusion:

• Function (i) is continuous.
• Function (ii) is continuous.
• Function (iii) is not continuous.

Therefore, the correct answer is (F) (i) and (ii) only.

Would you like further details on any of these calculations? Here are 5 related questions for further exploration:

1. What are the conditions for a function to be continuous at a point?
2. How can we handle discontinuities in piecewise functions?
3. Why is logarithmic function $\ln x$ undefined for $x \leq 0$?
4. What is the significance of removable discontinuities, like in function (iii)?
5. Can a function be continuous but not differentiable? If so, how?

Tip: When checking the continuity of piecewise functions, always check if the limits from both sides are equal at the point where the pieces join!