Resolved: "Use the method of your choice to evaluate..."

Zoe Oneal

Answered question

2021-01-02

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

lim_{(x, y) \to (0, 1)} \frac{y \sin x}{x (y + 1)}

aprovard

Skilled2021-01-03Added 94 answers

Given limits:

lim_{(x, y) \to (0, 1)} \frac{y \sin x}{x (y + 1)}

We have to evaluate the limit.
The limit is computed as follows:

lim_{(x, y) \to (0, 1)} \frac{y \sin x}{x (y + 1)} = lim_{x \to 0} \frac{\sin x}{x} lim_{y \to 1} \frac{y}{y + 1}

Substitutes the limits:
We know that:

lim_{θ \to 0} \frac{\sin θ}{θ} = 1

lim_{(x, y) \to (0, 1)} \frac{y \sin x}{x (y + 1)} = 1 \times \frac{1}{1 + 1}

lim_{(x, y) \to (0, 1)} \frac{y \sin x}{x (y + 1)} = \frac{1}{2}

Therefore, the value of

lim_{(x, y) \to (0, 1)} \frac{y \sin x}{x (y + 1)}

\frac{1}{2}

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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