Let's solve the problem step by step.
Given the limit:
[tex]\[
\lim_{x \rightarrow 4} \frac{40 - 4x - \left|x^2 - 10x\right|}{\left|x^2 - 100\right| - 84}
\][/tex]
### Step 1: Handling the absolute values
First, we need to deal with the absolute values. Note that the expressions inside absolute values change their behavior depending on the value of [tex]\(x\)[/tex].
#### Numerator: [tex]\(40 - 4x - |x^2 - 10x|\)[/tex]
To understand how [tex]\( |x^2 - 10x| \)[/tex] behaves, we analyze the quadratic [tex]\( x^2 - 10x \)[/tex]:
[tex]\[
x^2 - 10x = x(x - 10)
\][/tex]
- For [tex]\(x < 0\)[/tex] or [tex]\(x > 10\)[/tex], [tex]\(x(x - 10) \geq 0\)[/tex].
- For [tex]\(0 \leq x \leq 10\)[/tex], [tex]\(x(x - 10) \leq 0\)[/tex].
Given that [tex]\(x \rightarrow 4\)[/tex], we fall in the range [tex]\(0 \leq x \leq 10\)[/tex]. Thus, [tex]\( |x^2 - 10x| = -(x^2 - 10x) = 10x - x^2 \)[/tex].
Thus, the numerator becomes:
[tex]\[
40 - 4x - (10x - x^2) = 40 - 4x - 10x + x^2 = x^2 - 14x + 40
\][/tex]
#### Denominator: [tex]\( |x^2 - 100| - 84 \)[/tex]
Next, let's analyze the quadratic [tex]\( x^2 - 100 \)[/tex]:
[tex]\[
x^2 - 100 = (x - 10)(x + 10)
\][/tex]
- For [tex]\(x > 10\)[/tex] or [tex]\(x < -10\)[/tex], [tex]\(x^2 - 100 \geq 0\)[/tex].
- For [tex]\(-10 \leq x \leq 10\)[/tex], [tex]\(x^2 - 100 \leq 0\)[/tex].
Given that [tex]\(x \rightarrow 4\)[/tex], we fall in the range [tex]\(-10 \leq x \leq 10\)[/tex]. Thus, [tex]\( |x^2 - 100| = -(x^2 - 100) = 100 - x^2 \)[/tex].
Thus, the denominator becomes:
[tex]\[
|x^2 - 100| - 84 = (100 - x^2) - 84 = 100 - x^2 - 84 = 16 - x^2
\][/tex]
### Step 2: Simplifying the limit
Now, we have:
[tex]\[
\lim_{x \rightarrow 4} \frac{x^2 - 14x + 40}{16 - x^2}
\][/tex]
### Step 3: Evaluate the limit
Substitute [tex]\(x = 4\)[/tex] directly into the simplified expression:
[tex]\[
\frac{4^2 - 14 \cdot 4 + 40}{16 - 4^2} = \frac{16 - 56 + 40}{16 - 16} = \frac{0}{0}
\][/tex]
This indicates an indeterminate form [tex]\( \frac{0}{0} \)[/tex]. To resolve this, we factor both the numerator and the denominator:
Numerator:
[tex]\[
x^2 - 14x + 40
\][/tex]
Let's factorize [tex]\(x^2 - 14x + 40\)[/tex]. We look for two numbers that multiply to 40 and add up to -14. These numbers are -4 and -10:
[tex]\[
x^2 - 14x + 40 = (x - 4)(x - 10)
\][/tex]
Denominator:
[tex]\[
16 - x^2 = -(x^2 - 16) = -(x - 4)(x + 4)
\][/tex]
Thus, our limit expression becomes:
[tex]\[
\lim_{x \rightarrow 4} \frac{(x - 4)(x - 10)}{-(x - 4)(x + 4)}
\][/tex]
### Step 4: Cancel common factors and evaluate
We cancel the common factor [tex]\((x - 4)\)[/tex] from the numerator and the denominator:
[tex]\[
\lim_{x \rightarrow 4} \frac{x - 10}{-(x + 4)}
\][/tex]
Evaluate the limit by substituting [tex]\(x = 4\)[/tex]:
[tex]\[
\frac{4 - 10}{-(4 + 4)} = \frac{-6}{-8} = \frac{6}{8} = \frac{3}{4}
\][/tex]
### Final Answer
Therefore, the value of the limit is:
[tex]\[
\boxed{\frac{3}{4}}
\][/tex]
