Evaluate the following Limits
`lim_(xto 1)(2x^(8)-3x^(2)+1)/(x^(8)+6x^(5)-7)`

Answer

To evaluate the limit

\[ \lim_{x \to 1} \frac{2x^8 - 3x^2 + 1}{x^8 + 6x^5 - 7}, \]

we will follow these steps:

### Step 1: Direct Substitution
First, we substitute \( x = 1 \) directly into the expression:

\[ \text{Numerator: } 2(1)^8 - 3(1)^2 + 1 = 2 - 3 + 1 = 0, \]
\[ \text{Denominator: } (1)^8 + 6(1)^5 - 7 = 1 + 6 - 7 = 0. \]

Since both the numerator and denominator evaluate to 0, we have an indeterminate form \( \frac{0}{0} \).

**Hint:** When you encounter a \( \frac{0}{0} \) form, consider using L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we differentiate the numerator and the denominator separately.

**Numerator:**
\[ \frac{d}{dx}(2x^8 - 3x^2 + 1) = 16x^7 - 6x. \]

**Denominator:**
\[ \frac{d}{dx}(x^8 + 6x^5 - 7) = 8x^7 + 30x^4. \]

### Step 3: Rewrite the Limit
Now we rewrite the limit using the derivatives:

\[ \lim_{x \to 1} \frac{16x^7 - 6x}{8x^7 + 30x^4}. \]

### Step 4: Substitute \( x = 1 \) Again
Now we substitute \( x = 1 \) into the new expression:

\[ \text{Numerator: } 16(1)^7 - 6(1) = 16 - 6 = 10, \]
\[ \text{Denominator: } 8(1)^7 + 30(1)^4 = 8 + 30 = 38. \]

### Step 5: Calculate the Limit
Now we can evaluate the limit:

\[ \lim_{x \to 1} \frac{10}{38} = \frac{5}{19}. \]

Thus, the final answer is:

\[ \frac{5}{19}. \]

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