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Answer :
The limit of [tex]√(4 - x^2)[/tex] as x approaches 2 from the left side is 0, the expression under the square root, [tex]4 - x^2[/tex], is non-negative for x values less than 2, the square root function is defined.
1) The limit of [tex]e^x[/tex] as x approaches negative infinity can be found by considering the behavior of the exponential function as x becomes increasingly negative. Since [tex]e^x[/tex] is an increasing function, it will approach positive infinity as x approaches negative infinity. So, the limit is positive infinity.
2) For the function f(x) given by f(x) = {[tex]x^2[/tex] - 4x if x ≠ 3; 4x - 15 if x = 3}, we can find the limit as x approaches 3 by evaluating the left-hand and right-hand limits separately.
Considering the left-hand limit, as x approaches 3 from the left side (x < 3), the function becomes [tex]x^2 - 4x[/tex]. Plugging in x = 3, we get[tex](3)^2[/tex] - 4(3) = 9 - 12 = -3.
Considering the right-hand limit, as x approaches 3 from the right side (x > 3), the function becomes 4x - 15. Plugging in x = 3, we get 4(3) - 15 = 12 - 15 = -3.
Since the left-hand limit and right-hand limit are both -3, the limit of f(x) as x approaches 3 is -3.
3) The limit of [tex]√(4 - x^2)[/tex] as x approaches 2 from the left side (x < 2) can be found by considering the behavior of the square root function.
Since the expression under the square root, [tex]4 - x^2[/tex], is non-negative for x values less than 2, the square root function is defined.
Plugging in x = 2, we get [tex]√(4 - 2^2) = √(4 - 4) = √0 = 0.[/tex]
Therefore, the limit of [tex]√(4 - x^2)[/tex] as x approaches 2 from the left side is 0.
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