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sp2606 [1]
3 years ago
14

Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim (x, y) → (0, 0) x4 − 34y2 x2 + 17y2

Mathematics
1 answer:
HACTEHA [7]3 years ago
4 0

Answer:

<h2>DNE</h2>

Step-by-step explanation:

Given the limit of the function \lim_{(x,y) \to (0,0)} \frac{x^4-34y^2}{x^2+17y^2}, to find the limit, the following steps must be taken.

Step 1: Substitute the limit at x = 0 and y = 0 into the function

= \lim_{(x,y) \to (0,0)} \frac{x^4-34y^2}{x^2+17y^2}\\=  \frac{0^4-34(0)^2}{0^2+17(0)^2}\\= \frac{0}{0} (indeterminate)

Step 2: Substitute y = mx int o the function and simplify

= \lim_{(x,mx) \to (0,0)} \frac{x^4-34(mx)^2}{x^2+17(mx)^2}\\\\= \lim_{(x,mx) \to (0,0)} \frac{x^4-34m^2x^2}{x^2+17m^2x^2}\\\\\\= \lim_{(x,mx) \to (0,0)} \frac{x^2(x^2-34m^2)}{x^2(1+17m^2)}\\\\\\= \lim_{(x,mx) \to (0,0)} \frac{x^2-34m^2}{1+17m^2}\\

= \frac{0^2-34m^2}{1+17m^2}\\\\=  \frac{34m^2}{1+17m^2}\\\\

<em>Since there are still variable 'm' in the resulting function, this shows that the limit of the function does not exist, Hence, the function DNE</em>

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