Problem 13 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(-3,3)}\left(4 x^{2}-y^{2}\right)$$

Step-by-Step Solution

Answer

Answer: The limit is 27.

Step 1: Identify the Limit

The given problem is to evaluate the limit as \((x, y)\) approaches \((-3, 3)\) for the function: $$\lim _{(x, y) \rightarrow(-3,3)}\left(4 x^{2}-y^{2}\right)$$

Step 2: Substitute the Values for \(x\) and \(y\)

We will now substitute the values of \(x = -3\) and \(y = 3\). This will help us to evaluate the limit: $$\lim _{(x, y) \rightarrow(-3,3)}\left(4 (-3)^{2}-(3)^{2}\right)$$

Step 3: Compute the Expression

Now, we can compute the values: $$\lim _{(x, y) \rightarrow(-3,3)}\left(4 (9)-(9)\right) = \lim _{(x, y) \rightarrow(-3,3)}\left(36-9\right)$$

Step 4: Find the Result

Finally, compute the remaining values: $$\lim _{(x, y) \rightarrow(-3,3)}\left(27\right) = 27$$ So, the given limit is equal to 27.