8. Use the precise (formal definition) of the limit to prove that 8.1. $$ \lim _{x \rightarrow 3}(2 x-3)=3 $$ 8.2. $$ \lim _{x \rightarrow -3}(2 x-3)=-9 $$ 8.3. $$ \lim _{x \rightarrow 2} x^{2}=4 $$

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To prove that $$ \lim _{x \rightarrow 3}(2 x-3)=3 $$ using the formal definition of a limit, we need to show that for every $$ \epsilon > 0 $$, there exists a $$ \delta > 0 $$ such that if $$ 0 < |x - 3| < \delta $$, then $$ |(2x - 3) - 3| < \epsilon $$. Starting with the expression: $$ |(2x - 3) - 3| = |2x - 6| = 2|x - 3| $$ We want: $$ 2|x - 3| < \epsilon $$ This implies: $$ |x - 3| < \frac{\epsilon}{2} $$ Thus, we can choose $$ \delta = \frac{\epsilon}{2} $$. Therefore, we have shown that $$ \lim _{x \rightarrow 3}(2 x-3)=3 $$. Next, we prove that $$ \lim _{x \rightarrow -3}(2 x-3)=-9 $$ using the same formal definition. We need to show that for every $$ \epsilon > 0 $$, there exists a $$ \delta > 0 $$ such that if $$ 0 < |x + 3| < \delta $$, then $$ |(2x - 3) + 9| < \epsilon $$. Starting with the expression: $$ |(2x - 3) + 9| = |2x + 6| = 2|x + 3| $$ We want: $$ 2|x + 3| < \epsilon $$ This implies: $$ |x + 3| < \frac{\epsilon}{2} $$ Thus, we can choose $$ \delta = \frac{\epsilon}{2} $$. Therefore, we have shown that $$ \lim _{x \rightarrow -3}(2 x-3)=-9 $$. Finally, we prove that $$ \lim _{x \rightarrow 2} x^{2}=4 $$ using the formal definition. We need to show that for every $$ \epsilon > 0 $$, there exists a $$ \delta > 0 $$ such that if $$ 0 < |x - 2| < \delta $$, then $$ |x^{2} - 4| < \epsilon $$. Starting with the expression: $$ |x^{2} - 4| = |(x - 2)(x + 2)| $$ To control this, we can bound $$ |x + 2| $$. If we restrict $$ |x - 2| < 1 $$, then $$ 1 < x < 3 $$ implies $$ 3 < x + 2 < 5 $$, so $$ |x + 2| < 5 $$. Thus, we have: $$ |(x - 2)(x + 2)| < 5|x - 2| $$ We want: $$ 5|x - 2| < \epsilon $$ This implies: $$ |x - 2| < \frac{\epsilon}{5} $$ Thus, we can choose $$ \delta = \min\left(1, \frac{\epsilon}{5}\right) $$. Therefore, we have shown that $$ \lim _{x \rightarrow 2} x^{2}=4 $$. We have proven all three limits using the formal definition of limits: 1. $$ \lim _{x \rightarrow 3}(2 x-3)=3 $$ 2. $$ \lim _{x \rightarrow -3}(2 x-3)=-9 $$ 3. $$ \lim _{x \rightarrow 2} x^{2}=4 $$

Use the precise (formal definition) of the limit to prove that
8.1.
8.2.
8.3.

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Use the precise (formal definition) of the limit to prove that 8.1. 8.2. 8.3.