Final answer:
The question seems to ask about finding limits of a function as the point approaches (1, 3), but it is not clearly stated. If we assume a function, you can substitute the values directly or use alternative techniques if the direct substitution results in an indeterminate form. Plot key points and behavior when graphing functions on the same diagram.
Step-by-step explanation:
The question asks to find the limit of a function as the point (x, y) approaches (1, 3). The function is not given directly, but it seems to involve a quotient that has a quadratic expression in terms of x and y in the numerator and a linear expression in the denominator. To find the limit, we would generally substitute the point directly into the function. However, if this results in an indeterminate form, then we might need to use techniques such as factoring, l'Hôpital's rule, or approaching the limit along different paths to verify its existence and value.
To demonstrate the procedure, let's assume the function is correctly written as f(x, y) = (y^2 − 9x^2)/(3x − y). To find lim (x,y) → (1,3) f(x, y), we can substitute the values of x and y into the function:
- Substitute x = 1 and y = 3 into the function
- Simplify the expressions in the numerator and the denominator
- If the function is continuous at (1, 3), the direct substitution gives the limit
If the limit does not exist or if the substitution gives an indeterminate form like 0/0, we may need to simplify the expression further or consider other methods to evaluate the limit.
For sketching graphs like y = x, y = (x - 2)^2 - 7, y = e^x, and y = -e^{-x} on the same diagram, we would plot each equation on a graph, paying close attention to key points, intercepts, and asymptotic behavior.