Answer:
Step-by-step explanation:
Given the limit of the function [tex]\lim_{(x,y) \to (0,0)} \frac{x^4-34y^2}{x^2+17y^2}[/tex], to find the limit, the following steps must be taken.
Step 1: Substitute the limit at x = 0 and y = 0 into the function
[tex]= \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)[/tex]
Step 2: Substitute y = mx int o the function and simplify
[tex]= \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}\\[/tex]
[tex]= \frac{0^2-34m^2}{1+17m^2}\\\\= \frac{34m^2}{1+17m^2}\\\\[/tex]
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
