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

Let's analyze the function \( f(x) = \frac{2x + 12}{x + 6} \) as \( x \) approaches 6. ### Simplifying the Function First, simplify \( f(x) \): \[ f(x) = \frac{2x + 12}{x + 6} = \frac{2(x + 6)}{x + 6} \] For all \( x \neq -6 \), we can cancel out \( x + 6 \): \[ f(x) = 2 \] This simplification shows that \( f(x) \) is a constant function equal to 2 for all \( x \) except \( x = -6 \), where the function is undefined. ### (i) \( \lim_{x \to 6^{+}} f(x) \) As \( x \) approaches 6 from the right (\( x \to 6^{+} \)), the function \( f(x) \) remains: \[ f(x) = 2 \] Thus, \[ \lim_{x \to 6^{+}} f(x) = 2 \] ### (ii) \( \lim_{x \to 6^{-}} f(x) \) Similarly, as \( x \) approaches 6 from the left (\( x \to 6^{-} \)), the function \( f(x) \) is still: \[ f(x) = 2 \] So, \[ \lim_{x \to 6^{-}} f(x) = 2 \] ### (iii) Deduce \( \lim_{x \to 6} f(x) \) Since both the right-hand limit and the left-hand limit as \( x \) approaches 6 are equal to 2, the overall limit exists and is equal to 2. \[ \lim_{x \to 6} f(x) = 2 \] **Final Answer:** All three limits are equal to two. Specifically, (i) 2, (ii) 2, and therefore (iii) limₓ→6 f (x ) = 2.