> ƇαƖƇυƖαтє тнє Ɩιмιт σƑ (3x^3 - 2x^2 + 5x - 7) / (x^2 - 4) αѕ x αρρяσαƇнєѕ 2....
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> ƇαƖƇυƖαтє тнє Ɩιмιт σƑ (3x^3 - 2x^2 + 5x - 7) / (x^2 - 4) αѕ x αρρяσαƇнєѕ 2....
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To calculate the limit of [tex](3x^3 - 2x^2 + 5x - 7) / (x^2 - 4) [/tex]as [tex](x) [/tex]approaches 2, we can directly substitute \(x = 2\) into the expression:
[tex]\[\lim_{{x \to 2}} \frac{{3x^3 - 2x^2 + 5x - 7}}{{x^2 - 4}} = \frac{{3(2)^3 - 2(2)^2 + 5(2) - 7}}{{(2)^2 - 4}}\][/tex]
Simplifying the expression:
[tex][\frac{{3(8) - 2(4) + 10 - 7}}{{4 - 4}} = \frac{{24 - 8 + 10 - 7}}{{0}}][/tex]
This expression results in an indeterminate form [tex](\frac{0}{0})[/tex], indicating that further simplification is needed. We can factor the numerator and denominator:
[tex][\frac{{(3x - 1)(x^2 + x - 7)}}{{(x - 2)(x + 2)}}][/tex]
Now, the common factor of \((x - 2)\) cancels out:
[tex][\frac{{3x - 1}}{{x + 2}}][/tex]
Now, substitute \(x = 2\) into this simplified expression:
[tex]\[\frac{{3(2) - 1}}{{2 + 2}} = \frac{{5}}{{4}}\][/tex]
Therefore, the limit of [tex]((3x^3 - 2x^2 + 5x - 7) / (x^2 - 4)[/tex] as [tex](x)[/tex] approaches 2 is[tex](\frac{5}{4}).[/tex]
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ƇαƖƇυƖαтє тнє Ɩιмιт σƑ (3x^3 - 2x^2 + 5x - 7) / (x^2 - 4) αѕ x αρρяσαƇнєѕ 2....
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