Evaluate the following limits in Exercise `lim_(x to -1)(x^(10)+x^(5)+1)/(x-1)`

Answer

To evaluate the limit \( \lim_{x \to -1} \frac{x^{10} + x^5 + 1}{x - 1} \), we will follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = -1 \) into the expression to check if the limit is defined.

\[ \text{Numerator: } (-1)^{10} + (-1)^5 + 1 = 1 - 1 + 1 = 1 \]
\[ \text{Denominator: } -1 - 1 = -2 \]

So, we have:
\[ \frac{1}{-2} = -\frac{1}{2} \]

### Step 2: Conclusion
Since substituting \( x = -1 \) gives us a defined value, we conclude that:
\[ \lim_{x \to -1} \frac{x^{10} + x^5 + 1}{x - 1} = -\frac{1}{2} \]

### Final Answer
Thus, the limit is:
\[ -\frac{1}{2} \]

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