"Find the limit, if it exists, or show..." was answered by students like you

suffisantfn

Open question

2022-08-20

Find the limit, if it exists, or show that the limit does not exist.

lim (x, y)

tends to (1,0)

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

Carmelo Peck

Beginner2022-08-21Added 12 answers

For the limit

lim_{(x, y) \Rightarrow (1, 0)} \frac{x y - y}{{(x - 1)}^{2} + y^{2}}

We will replace y with m(x-1), to get

lim_{x \Rightarrow 1} \frac{x \cdot m (x - 1) - m (x - 1)}{{(x - 1)}^{2} + m^{2} {(x - 1)}^{2}}

We can cancel (x-1) from the numerator and the denominator

= lim_{x \Rightarrow 1} \frac{x \cdot m - m}{(x - 1) + m^{2} (x - 1)}

= lim_{x \Rightarrow 1} \frac{m (x - 1)}{(x - 1) + m^{2} (x - 1)}

We can again cancel (x-1) from the numerator and the denominator

- lim_{x \Rightarrow 1} \frac{m}{1 + m^{2}} = \frac{m}{1 + m^{2}}

Since the limit is not independent of m, it does not exist
Result:
The limit does not exist

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