Question 5 of 70beginner🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→−1x10+x5+1x−1\lim_{x \to -1} \frac{x^{10} + x^5 + 1}{x - 1}

Correct Answer

-1/2

Exercise: EXERCISE 12.1 | Q: 5 | (Page Page 21)
For More Understanding

Explanation

The given function is a rational function. According to the approach shown in the textbook, for rational functions we first evaluate the function at the given point. If the denominator is non-zero, the limit is simply the value of the function at that point. Here, at x = -1, the denominator is -2 (non-zero), so direct substitution is valid.

Solution Steps

  1. Step 1: Recognise that the function is rational and the denominator is not zero at x = -1.

  2. Step 2: Substitute x = -1 directly into the expression.

  3. Step 3: Compute numerator: (-1)^10 + (-1)^5 + 1 = 1 + (-1) + 1 = 1.

  4. Step 4: Compute denominator: -1 - 1 = -2.

  5. Step 5: Therefore, the limit equals 1 / (-2) = -1/2.

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