Expert answers to "Use the method of your choice to evaluate..."

Clifland

Answered question

2021-01-10

Use the method of your choice to evaluate the following limits.

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{2 x^{2} - x y - y^{2}}

Dora

Skilled2021-01-11Added 98 answers

Given:

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{2 x^{2} - x y - y^{2}}

To evaluate:
The given limit.
Solution:
Here,

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{2 x^{2} - x y - y^{2}}

On simplifying the equation,
The required equation is,

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{x^{2} - x y 2 - y^{2}} = lim_{(x, y) \to (1, 1)} \frac{x + 2 y}{2 x + y}

Plug in the values (x,y)=(1,1)

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{x^{2} - x y 2 - y^{2}} = \frac{1 + 2 \times 1}{2 \times 1 + 1}

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{x^{2} - x y 2 - y^{2}} = \frac{1 + 2}{2 + 1}

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{x^{2} - x y 2 - y^{2}} = \frac{3}{3}

lim_{(x, y) \to (1, 1)} \frac{x^{2} + x y - 2 y^{2}}{x^{2} - x y 2 - y^{2}} = 1

Which is required.

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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