3. Let $$ G(x)=\left\{\begin{array}{ll}4-\frac{1}{2} x & ext { if } x2 .\end{array} ight. $$ Find the value of $$ \eta $$ so that $$ \lim _{x ightarrow 2} G(x) $$ exists.

Question

3. Let $$ G(x)=\left\{\begin{array}{ll}4-\frac{1}{2} x & 	ext { if } x<2 \ \sqrt{x+\eta} & 	ext { if } x>2 .\end{array}ight. $$ Find the value of $$ \eta $$ so that $$ \lim _{x ightarrow 2} G(x) $$ exists.

UpStudy AI · Accepted Answer

1. For $$ x < 2 $$, the function is defined as
   $$
   G(x)=4-\frac{1}{2} x.
   $$
   The left-hand limit as $$ x $$ approaches 2 is
   $$
   \lim_{x \to 2^-} G(x)=4-\frac{1}{2}(2)=4-1=3.
   $$

2. For $$ x > 2 $$, the function is defined as
   $$
   G(x)=\sqrt{x+\eta}.
   $$
   The right-hand limit as $$ x $$ approaches 2 is
   $$
   \lim_{x \to 2^+} G(x)=\sqrt{2+\eta}.
   $$

3. For the overall limit $$ \lim_{x \rightarrow 2} G(x) $$ to exist, the left-hand and right-hand limits must be equal. Therefore, we set
   $$
   \sqrt{2+\eta}=3.
   $$

4. Squaring both sides gives
   $$
   2+\eta=9.
   $$

5. Solving for $$ \eta $$:
   $$
   \eta=9-2=7.
   $$

Thus, the value of $$ \eta $$ is $$ 7 $$.
To ensure the limit $$ \lim_{x \to 2} G(x) $$ exists, set the left-hand limit equal to the right-hand limit: $$ \sqrt{2 + \eta} = 3 $$ Squaring both sides: $$ 2 + \eta = 9 $$ Solving for $$ \eta $$: $$ \eta = 7 $$ So, $$ \eta = 7 $$.

Let Find the value of so that exists.

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