Question

Question 4 (6 marks) (1) Evaluate limy→2 limx-2 19-√2x+5-x-2 X-2 19-2x+5-4 x-2 (Hint: Rationalize twice) (ii) Use the result of part (1) to evaluate (Hint: It's a one line solution using part (1).)

Answer 1

`\[ \lim_{{y \to 2}} \lim_{{x \to 2}} (19 - √(2x + 5) - x + 2)/(19 - 2x + 5 - 4) = 1 \]`

Step-by-step explanation:

**(i) Evaluate
`\(\lim_{{y \to 2}} \lim_{{x \to 2}} (19 - √(2x + 5) - x + 2)/(19 - 2x + 5 - 4)\)`:**

To find this limit, we can start by rationalizing the numerator and denominator twice.

First, rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the expression involving the square root:

`\[ \lim_{{y \to 2}} \lim_{{x \to 2}} ((19 - √(2x + 5) - x + 2)(19 + √(2x + 5) + x - 2))/((19 - 2x + 5 - 4)(19 + √(2x + 5) + x - 2)) \]`

Simplify the expression, and rationalize the denominator again by multiplying by the conjugate:

`\[ \lim_{{y \to 2}} \lim_{{x \to 2}} (-(2x + 5 - x + 2))/(-2) \]`

Simplify further:

`\[ \lim_{{y \to 2}} \lim_{{x \to 2}} (-x)/(-2) \]`

Now, the limit as
`\(x\)` approaches 2 is:

`\[ \lim_{{y \to 2}} (-2)/(-2) = 1 \]`

**(ii) Use the result of part (i) to evaluate:**

The result of part (i) is 1, so:

`\[ \text{The final result is } 1 \]`