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Calculate the following limits:

1. Simplify and calculate:
[tex]\[ \frac{x^2-36}{x-6} \][/tex]

2. Calculate the limit:
[tex]\[ \lim_{x \rightarrow 0} \frac{16x - 8x}{4x^2 + 2x} \][/tex]
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This revised version makes logical sense based on common mathematical problems involving simplification and limits.



Answer :

Let's solve the following two problems step-by-step.

### Problem 1:
Simplify the expression [tex]\(\frac{x^2 - 36}{x - 6}\)[/tex]

Solution:

First, notice that the numerator [tex]\(x^2 - 36\)[/tex] can be factored using the difference of squares formula: [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].

In this case, we have [tex]\( x^2 - 36 = (x - 6)(x + 6) \)[/tex].

So we can rewrite the expression as:

[tex]\[ \frac{x^2 - 36}{x - 6} = \frac{(x - 6)(x + 6)}{x - 6} \][/tex]

If [tex]\(x \neq 6\)[/tex], the [tex]\( x - 6 \)[/tex] terms cancel each other out, leaving:

[tex]\[ \frac{(x - 6)(x + 6)}{x - 6} = x + 6 \][/tex]

Therefore,

[tex]\[ \frac{x^2 - 36}{x - 6} = x + 6 \quad \text{for} \; x \neq 6 \][/tex]

### Problem 2:
Calculate the limit [tex]\(\lim_{x \to 0} \frac{16x - 8x}{4x^2 + 2x}\)[/tex]

Solution:

First, simplify the expression inside the limit:

[tex]\[ \frac{16x - 8x}{4x^2 + 2x} \][/tex]

Combine terms in the numerator:

[tex]\[ \frac{8x}{4x^2 + 2x} \][/tex]

Factor out the common terms in both the numerator and the denominator. The numerator can be factored as:

[tex]\[ 8x \][/tex]

For the denominator:

[tex]\[ 4x^2 + 2x = 2x(2x + 1) \][/tex]

So, the expression becomes:

[tex]\[ \frac{8x}{2x(2x + 1)} \][/tex]

We can cancel out the common factor [tex]\(2x\)[/tex]:

[tex]\[ \frac{8x}{2x(2x + 1)} = \frac{8}{2(2x + 1)} = \frac{8}{4x + 2} \][/tex]

Now, take the limit as [tex]\(x\)[/tex] approaches 0:

[tex]\[ \lim_{x \to 0} \frac{8}{4x + 2} \][/tex]

Substitute [tex]\(x = 0\)[/tex]:

[tex]\[ \frac{8}{4(0) + 2} = \frac{8}{2} = 4 \][/tex]

Therefore,

[tex]\(\lim_{x \to 0} \frac{16x - 8x}{4x^2 + 2x} = 4\)[/tex]

By following these detailed steps, we reach the simplified expressions and limits for the given mathematical problems.