Let f(x)=x−2x2−4. <IMAGE> Make a conjecture about the value of x→2limx−2x2−4.
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First, recognize that the expression \( \frac{x^2 - 4}{x - 2} \) is undefined at \( x = 2 \) because it results in division by zero. Therefore, we need to simplify the expression to evaluate the limit.
Notice that the numerator \( x^2 - 4 \) can be factored as a difference of squares: \( x^2 - 4 = (x - 2)(x + 2) \).
Substitute the factored form into the original expression: \( \frac{(x - 2)(x + 2)}{x - 2} \).
Cancel the common factor \( (x - 2) \) in the numerator and the denominator, which simplifies the expression to \( x + 2 \), provided \( x \neq 2 \).
Now, evaluate the limit of the simplified expression as \( x \to 2 \): \( \lim_{x \to 2} (x + 2) \). This can be directly computed by substituting \( x = 2 \) into the expression, leading to the conjecture about the limit.
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