(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 $$ x=6 $$.

Question

UpStudy AI · Accepted Answer

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.
All three limits are equal to 2.

© Let
(i)
(ii)
(iii) Use (i) and (ii) to deduce at .

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions

© Let (i) (ii) (iii) Use (i) and (ii) to deduce at .

Upstudy AI Solution

Answer

Solution

Extra Insights

Related Questions

Latest Calculus Questions

© Let
(i)
(ii)
(iii) Use (i) and (ii) to deduce at .