Evaluate the limit Lim (x-2)/(x²-4)x→2
To evaluate the limit Lim (x-2)/(x²-4) as x approaches 2, we can try substituting x with 2 directly
To evaluate the limit Lim (x-2)/(x²-4) as x approaches 2, we can try substituting x with 2 directly. However, this would result in an undefined expression as it would involve division by zero. Instead, we need to find another approach to evaluate the limit.
The factorization of the denominator, x² – 4, can be done using the difference of squares identity, which states that a² – b² can be factored as (a – b)(a + b). Applying this identity, we have x² – 4 = (x – 2)(x + 2).
Now we can rewrite the expression as follows:
(x – 2)/(x² – 4) = (x – 2)/((x – 2)(x + 2))
Notice that the (x – 2) terms in the numerator and the denominator can now cancel each other out. This is possible because we are assuming x is not equal to 2.
Canceling out the common factors, we get:
1/(x + 2)
Now, we can substitute x with 2:
1/(2 + 2) = 1/4
Therefore, the limit Lim (x-2)/(x²-4) as x approaches 2 is equal to 1/4.
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