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Sagot :
To solve the limit [tex]\(\lim_{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4}\)[/tex], we will proceed as follows:
1. Expression Simplification:
Let's first factorize the numerator and the denominator.
- The numerator [tex]\(x^5 - a^5\)[/tex] can be factored using the difference of powers formula:
[tex]\[ x^5 - a^5 = (x - a)(x^4 + x^3 a + x^2 a^2 + x a^3 + a^4) \][/tex]
- The denominator [tex]\(x^4 - a^4\)[/tex] can also be factored using the difference of squares formula twice:
[tex]\[ x^4 - a^4 = (x^2)^2 - (a^2)^2 = (x^2 - a^2)(x^2 + a^2) \][/tex]
Further, we can factor [tex]\(x^2 - a^2\)[/tex] using the difference of squares formula:
[tex]\[ x^2 - a^2 = (x - a)(x + a) \][/tex]
So,
[tex]\[ x^4 - a^4 = (x - a)(x + a)(x^2 + a^2) \][/tex]
2. Substitute Factored Forms:
Substitute the factored forms back into the original limit:
[tex]\[ \lim_{x \rightarrow a} \frac{(x - a)(x^4 + x^3 a + x^2 a^2 + x a^3 + a^4)}{(x - a)(x + a)(x^2 + a^2)} \][/tex]
3. Cancel Common Factors:
Note that both the numerator and denominator have the common factor [tex]\((x - a)\)[/tex]. Since we are taking the limit as [tex]\(x \)[/tex] approaches [tex]\(a\)[/tex], [tex]\(x \neq a\)[/tex] (except exactly at the limit). Therefore, we can safely cancel the [tex]\((x - a)\)[/tex] term:
[tex]\[ \lim_{x \rightarrow a} \frac{x^4 + x^3 a + x^2 a^2 + x a^3 + a^4}{(x + a)(x^2 + a^2)} \][/tex]
4. Evaluate the Limit:
Now that the expression is simplified, we substitute [tex]\(x = a\)[/tex] into the remaining expression:
[tex]\[ \frac{a^4 + a^3 a + a^2 a^2 + a a^3 + a^4}{(a + a)(a^2 + a^2)} \][/tex]
Simplify each term:
[tex]\[ a^4 + a^4 + a^4 + a^4 + a^4 = 5a^4 \][/tex]
[tex]\[ (a + a)(a^2 + a^2) = 2a \cdot 2a^2 = 4a^3 \][/tex]
So the expression now is:
[tex]\[ \frac{5a^4}{4a^3} = \frac{5}{4}a \][/tex]
Thus, the limit is:
[tex]\[ \boxed{\frac{5a}{4}} \][/tex]
1. Expression Simplification:
Let's first factorize the numerator and the denominator.
- The numerator [tex]\(x^5 - a^5\)[/tex] can be factored using the difference of powers formula:
[tex]\[ x^5 - a^5 = (x - a)(x^4 + x^3 a + x^2 a^2 + x a^3 + a^4) \][/tex]
- The denominator [tex]\(x^4 - a^4\)[/tex] can also be factored using the difference of squares formula twice:
[tex]\[ x^4 - a^4 = (x^2)^2 - (a^2)^2 = (x^2 - a^2)(x^2 + a^2) \][/tex]
Further, we can factor [tex]\(x^2 - a^2\)[/tex] using the difference of squares formula:
[tex]\[ x^2 - a^2 = (x - a)(x + a) \][/tex]
So,
[tex]\[ x^4 - a^4 = (x - a)(x + a)(x^2 + a^2) \][/tex]
2. Substitute Factored Forms:
Substitute the factored forms back into the original limit:
[tex]\[ \lim_{x \rightarrow a} \frac{(x - a)(x^4 + x^3 a + x^2 a^2 + x a^3 + a^4)}{(x - a)(x + a)(x^2 + a^2)} \][/tex]
3. Cancel Common Factors:
Note that both the numerator and denominator have the common factor [tex]\((x - a)\)[/tex]. Since we are taking the limit as [tex]\(x \)[/tex] approaches [tex]\(a\)[/tex], [tex]\(x \neq a\)[/tex] (except exactly at the limit). Therefore, we can safely cancel the [tex]\((x - a)\)[/tex] term:
[tex]\[ \lim_{x \rightarrow a} \frac{x^4 + x^3 a + x^2 a^2 + x a^3 + a^4}{(x + a)(x^2 + a^2)} \][/tex]
4. Evaluate the Limit:
Now that the expression is simplified, we substitute [tex]\(x = a\)[/tex] into the remaining expression:
[tex]\[ \frac{a^4 + a^3 a + a^2 a^2 + a a^3 + a^4}{(a + a)(a^2 + a^2)} \][/tex]
Simplify each term:
[tex]\[ a^4 + a^4 + a^4 + a^4 + a^4 = 5a^4 \][/tex]
[tex]\[ (a + a)(a^2 + a^2) = 2a \cdot 2a^2 = 4a^3 \][/tex]
So the expression now is:
[tex]\[ \frac{5a^4}{4a^3} = \frac{5}{4}a \][/tex]
Thus, the limit is:
[tex]\[ \boxed{\frac{5a}{4}} \][/tex]
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