Question 12 of 70intermediate🔧 ApplyNumerical1 mark

Evaluate the following limits in Exercises 1 to 22. lim⁡x→−21x+12x+2\lim_{x \to -2} \frac{\frac{1}{x} + \frac{1}{2}}{x + 2}

Correct Answer

-14\frac{1}{4}

Exercise: EXERCISE 12.1 | Q: 12 | (Page Page 21)
For More Understanding

Explanation

This is a limit problem of the indeterminate form 0/0. Following the textbook approach shown in examples (ii), (iii), and (iv), we first simplify the expression algebraically before substituting the limit value. The key technique is to combine fractions in the numerator, then factor and cancel the common term (x+2) since x ≠\neq -2 when taking the limit.

Solution Steps

  1. Step 1: Evaluate at x = -2 to check the form. Numerator = 1/(-2) + 1/2 = 0, Denominator = -2 + 2 = 0. The function is of the form 0/0.

  2. Step 2: Simplify the numerator by combining fractions: (1/x + 1/2) = (2+x)/(2x).

  3. Step 3: The expression becomes [(2+x)/(2x)]/(x+2) = (2+x)/[2x(x+2)].

  4. Step 4: Cancel the common factor (x+2) since x ≠\neq -2: (x+2)/[2x(x+2)] = 1/(2x).

  5. Step 5: Evaluate the limit: lim(x→-2) 1/(2x) = 1/[2(-2)] = -1/4.

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