Solve for [tex]\[ x \][/tex].
[tex]\[ 3x = 6x - 2 \][/tex]

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[tex]\[ 9. \lim _{x \rightarrow a} \frac{x^5-a^5}{x^4-a^4} \][/tex]

Response:



Answer :

To solve the limit [tex]\(\lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4}\)[/tex], we need to consider the behavior of the given expression as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex]. Let's go through the solution step-by-step.

First, rewrite the given limit expression:

[tex]\[ \lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4} \][/tex]

Observe that both the numerator and the denominator become indeterminate forms [tex]\(0/0\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex]. To simplify this expression, we can factorize the numerator and denominator.

### Step 1: Factorize the Numerator and Denominator

The expression in the numerator [tex]\(x^5 - a^5\)[/tex] can be factored using the difference of powers:

[tex]\[ x^5 - a^5 = (x - a)(x^4 + x^3a + x^2a^2 + xa^3 + a^4) \][/tex]

Similarly, the expression in the denominator [tex]\(x^4 - a^4\)[/tex] can be factored using the difference of squares twice:

[tex]\[ x^4 - a^4 = (x^2 - a^2)(x^2 + a^2) = (x - a)(x + a)(x^2 + a^2) \][/tex]

Thus, we can rewrite the limit expression as:

[tex]\[ \frac{x^5 - a^5}{x^4 - a^4} = \frac{(x - a)(x^4 + x^3a + x^2a^2 + xa^3 + a^4)}{(x - a)(x + a)(x^2 + a^2)} \][/tex]

### Step 2: Cancel the Common Factor

Since [tex]\(x \neq a\)[/tex] (as we are just looking at the limit approaching [tex]\(a\)[/tex]), we can cancel out the common factor [tex]\((x - a)\)[/tex] in the numerator and the denominator:

[tex]\[ \frac{x^4 + x^3a + x^2a^2 + xa^3 + a^4}{(x + a)(x^2 + a^2)} \][/tex]

### Step 3: Evaluate the Limit as [tex]\(x\)[/tex] Approaches [tex]\(a\)[/tex]

Now, substitute [tex]\(x = a\)[/tex] in the simplified expression to evaluate the limit:

[tex]\[ \lim _{x \rightarrow a} \frac{x^4 + x^3a + x^2a^2 + xa^3 + a^4}{(x + a)(x^2 + a^2)} \][/tex]

Substituting [tex]\(x = a\)[/tex]:

[tex]\[ = \frac{a^4 + a^4 + a^4 + a^4 + a^4}{(a + a)(a^2 + a^2)} \][/tex]
[tex]\[ = \frac{5a^4}{2a \cdot 2a^2} \][/tex]
[tex]\[ = \frac{5a^4}{4a^3} \][/tex]
[tex]\[ = \frac{5a^4}{4a^3} = \frac{5a}{4} \][/tex]

So, the limit is:

[tex]\[ \lim _{x \rightarrow a} \frac{x^5 - a^5}{x^4 - a^4} = \frac{5a}{4} \][/tex]

Thus, the solution to the limit is:

[tex]\[ \boxed{\frac{5a}{4}} \][/tex]