Let $$ g(x)=\left\{\begin{array}{ll}x & ext { if } x

Question

Let $$ g(x)=\left\{\begin{array}{ll}x & 	ext { if } x<1 \ 3 & 	ext { if } x=1 \ 2-x^{2} & 	ext { if } 12\end{array}ight. $$ $$ \lim _{x ightarrow 2}[g(x)]=? $$

UpStudy AI · Accepted Answer

Evaluate the limit by following steps:
- step0: Evaluate the limit:
  $$\lim _{x\rightarrow 2}\left(gx\right)$$
- step1: Rewrite the expression:
  $$g\times \lim _{x\rightarrow 2}\left(x\right)$$
- step2: Calculate:
  $$g\times 2$$
- step3: Calculate:
  $$2g$$
To find $$ \lim_{x \rightarrow 2} g(x) $$, we need to evaluate the function $$ g(x) $$ as $$ x $$ approaches 2.

1. For $$ x < 1 $$, $$ g(x) = x $$.
2. For $$ x = 1 $$, $$ g(x) = 3 $$.
3. For $$ 1 < x \leq 2 $$, $$ g(x) = 2 - x^2 $$.
4. For $$ x > 2 $$, $$ g(x) = x - 3 $$.

Since we are interested in the limit as $$ x $$ approaches 2, we will consider the piece of the function that applies when $$ x $$ is just less than or equal to 2, which is $$ g(x) = 2 - x^2 $$.

Now, we can calculate the limit:

$$
\lim_{x \rightarrow 2} g(x) = 2 - (2)^2 = 2 - 4 = -2
$$

Thus, the limit is:

$$
\lim_{x \rightarrow 2} g(x) = -2
$$
The limit as $$ x $$ approaches 2 of $$ g(x) $$ is -2.

Let

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