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To determine the limit [tex]\(\operatorname{Lim}_{x \rightarrow 2} \frac{x-2}{x^2-4}\)[/tex], we need to evaluate the function [tex]\(\frac{x-2}{x^2-4}\)[/tex] as [tex]\(x\)[/tex] approaches 2. Let's first simplify the expression and then see what it converges to when [tex]\(x\)[/tex] approaches 2.
### Simplification
The function [tex]\(\frac{x-2}{x^2-4}\)[/tex] can be simplified as follows:
1. Notice that [tex]\(x^2 - 4\)[/tex] is a difference of squares:
[tex]\[ x^2 - 4 = (x - 2)(x + 2) \][/tex]
2. Thus, the given function can be rewritten as:
[tex]\[ \frac{x-2}{x^2-4} = \frac{x-2}{(x-2)(x+2)} \][/tex]
3. For [tex]\(x \neq 2\)[/tex], we can cancel out the common factor [tex]\((x - 2)\)[/tex]:
[tex]\[ \frac{x-2}{(x-2)(x+2)} = \frac{1}{x+2} \][/tex]
Now, our function is simplified to [tex]\(\frac{1}{x+2}\)[/tex] for [tex]\(x \neq 2\)[/tex].
### Evaluating the Limit
Next, we evaluate the limit of [tex]\(\frac{1}{x+2}\)[/tex] as [tex]\(x\)[/tex] approaches 2:
[tex]\[ \lim_{x \to 2} \frac{1}{x+2} \][/tex]
Since [tex]\(\frac{1}{x+2}\)[/tex] is continuous at [tex]\(x = 2\)[/tex], we can directly substitute [tex]\(x = 2\)[/tex] into the simplified function:
[tex]\[ \lim_{x \to 2} \frac{1}{x+2} = \frac{1}{2+2} = \frac{1}{4} \][/tex]
Therefore, the limit is:
[tex]\[ \boxed{\frac{1}{4}} \][/tex]
### Numerical Evaluation
To further illustrate this result, we can compute [tex]\(\frac{x-2}{x^2-4}\)[/tex] for values of [tex]\(x\)[/tex] close to 2, as given in the table.
1. When [tex]\(x = 1.9\)[/tex]:
[tex]\[ \frac{1.9-2}{1.9^2-4} = \frac{-0.1}{1.9^2-4} = \frac{-0.1}{3.61-4} = \frac{-0.1}{-0.39} \approx 0.2564 \][/tex]
2. When [tex]\(x = 1.99\)[/tex]:
[tex]\[ \frac{1.99-2}{1.99^2-4} \approx \frac{-0.01}{3.9601-4} = \frac{-0.01}{-0.0399} \approx 0.2506 \][/tex]
3. When [tex]\(x = 1.999\)[/tex]:
[tex]\[ \frac{1.999-2}{1.999^2-4} \approx \frac{-0.001}{3.996001-4} = \frac{-0.001}{-0.003999} \approx 0.2501 \][/tex]
4. When [tex]\(x = 2.000\)[/tex]:
Note: Here, the function is undefined at exactly [tex]\(x = 2\)[/tex] (division by zero).
5. When [tex]\(x = 2.001\)[/tex]:
[tex]\[ \frac{2.001-2}{2.001^2-4} \approx \frac{0.001}{4.004001-4} = \frac{0.001}{0.004001} \approx 0.2499 \][/tex]
6. When [tex]\(x = 2.01\)[/tex]:
[tex]\[ \frac{2.01-2}{2.01^2-4} \approx \frac{0.01}{4.0401-4} = \frac{0.01}{0.0401} \approx 0.2494 \][/tex]
7. When [tex]\(x = 2.1\)[/tex]:
[tex]\[ \frac{2.1-2}{2.1^2-4} = \frac{0.1}{4.41-4} = \frac{0.1}{0.41} \approx 0.2439 \][/tex]
These numerical values are approaching 0.25, consistent with the limit calculation:
[tex]\[ \boxed{\frac{1}{4}} \][/tex]
### Simplification
The function [tex]\(\frac{x-2}{x^2-4}\)[/tex] can be simplified as follows:
1. Notice that [tex]\(x^2 - 4\)[/tex] is a difference of squares:
[tex]\[ x^2 - 4 = (x - 2)(x + 2) \][/tex]
2. Thus, the given function can be rewritten as:
[tex]\[ \frac{x-2}{x^2-4} = \frac{x-2}{(x-2)(x+2)} \][/tex]
3. For [tex]\(x \neq 2\)[/tex], we can cancel out the common factor [tex]\((x - 2)\)[/tex]:
[tex]\[ \frac{x-2}{(x-2)(x+2)} = \frac{1}{x+2} \][/tex]
Now, our function is simplified to [tex]\(\frac{1}{x+2}\)[/tex] for [tex]\(x \neq 2\)[/tex].
### Evaluating the Limit
Next, we evaluate the limit of [tex]\(\frac{1}{x+2}\)[/tex] as [tex]\(x\)[/tex] approaches 2:
[tex]\[ \lim_{x \to 2} \frac{1}{x+2} \][/tex]
Since [tex]\(\frac{1}{x+2}\)[/tex] is continuous at [tex]\(x = 2\)[/tex], we can directly substitute [tex]\(x = 2\)[/tex] into the simplified function:
[tex]\[ \lim_{x \to 2} \frac{1}{x+2} = \frac{1}{2+2} = \frac{1}{4} \][/tex]
Therefore, the limit is:
[tex]\[ \boxed{\frac{1}{4}} \][/tex]
### Numerical Evaluation
To further illustrate this result, we can compute [tex]\(\frac{x-2}{x^2-4}\)[/tex] for values of [tex]\(x\)[/tex] close to 2, as given in the table.
1. When [tex]\(x = 1.9\)[/tex]:
[tex]\[ \frac{1.9-2}{1.9^2-4} = \frac{-0.1}{1.9^2-4} = \frac{-0.1}{3.61-4} = \frac{-0.1}{-0.39} \approx 0.2564 \][/tex]
2. When [tex]\(x = 1.99\)[/tex]:
[tex]\[ \frac{1.99-2}{1.99^2-4} \approx \frac{-0.01}{3.9601-4} = \frac{-0.01}{-0.0399} \approx 0.2506 \][/tex]
3. When [tex]\(x = 1.999\)[/tex]:
[tex]\[ \frac{1.999-2}{1.999^2-4} \approx \frac{-0.001}{3.996001-4} = \frac{-0.001}{-0.003999} \approx 0.2501 \][/tex]
4. When [tex]\(x = 2.000\)[/tex]:
Note: Here, the function is undefined at exactly [tex]\(x = 2\)[/tex] (division by zero).
5. When [tex]\(x = 2.001\)[/tex]:
[tex]\[ \frac{2.001-2}{2.001^2-4} \approx \frac{0.001}{4.004001-4} = \frac{0.001}{0.004001} \approx 0.2499 \][/tex]
6. When [tex]\(x = 2.01\)[/tex]:
[tex]\[ \frac{2.01-2}{2.01^2-4} \approx \frac{0.01}{4.0401-4} = \frac{0.01}{0.0401} \approx 0.2494 \][/tex]
7. When [tex]\(x = 2.1\)[/tex]:
[tex]\[ \frac{2.1-2}{2.1^2-4} = \frac{0.1}{4.41-4} = \frac{0.1}{0.41} \approx 0.2439 \][/tex]
These numerical values are approaching 0.25, consistent with the limit calculation:
[tex]\[ \boxed{\frac{1}{4}} \][/tex]
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