Evaluating the Limit of (x – 2)/(x^2 – 4) as x Approaches 2: Simplification and Algebraic Manipulations

Evaluate the limit Lim (x-2)/(x²-4)x→2

To evaluate the limit of (x – 2)/(x^2 – 4) as x approaches 2, we can try plugging in the value of 2 directly and see if it gives us a meaningful result

To evaluate the limit of (x – 2)/(x^2 – 4) as x approaches 2, we can try plugging in the value of 2 directly and see if it gives us a meaningful result. However, this will result in an indeterminate form of 0/0, which is undefined.

In such cases, we can simplify the expression or apply algebraic manipulations to rewrite it in a form that allows for further evaluation.

First, notice that both the numerator and denominator can be factored as the difference of squares:

x – 2 = (x – 2)
x^2 – 4 = (x – 2)(x + 2)

Canceling out the common factor of (x – 2), we are left with:

(x – 2)/(x^2 – 4) = 1/(x + 2)

Now we can attempt to find the limit:

Lim (x – 2)/(x^2 – 4) as x approaches 2
= Lim 1/(x + 2) as x approaches 2

By plugging in the value of 2 into the expression, we find:

Lim 1/(2 + 2) = Lim 1/4
= 1/4

Thus, the limit of (x – 2)/(x^2 – 4) as x approaches 2 is equal to 1/4.

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