Find the limit, if it exists, or type dne if it does not exist.lim(x,y)→(−35,5)xycos(x+7y)=

To find the limit $\lim_{(x,y) \to (-35,5)} xy \cos(x + 7y)$, we can follow these steps:

1. Break Down the Problem

We need to evaluate the limit of the function $f(x,y) = xy \cos(x + 7y)$ as $(x,y)$ approaches $(-35, 5)$.

2. Relevant Concepts

We generally apply the limit as follows:

• Evaluate $f(x,y) = xy \cos(x + 7y)$
• Handle the components separately as $(x,y)$ approaches the specified point.

3. Analysis and Detail

Evaluate the individual components at the limit point $(-35, 5)$:

• Calculate $x$ and $y$: $x \to -35, \quad y \to 5$

• Calculate $xy$: $xy = (-35)(5) = -175$

• Calculate $x + 7y$: $x + 7y = -35 + 7(5) = -35 + 35 = 0$

• Calculate $\cos(x + 7y)$: $\cos(0) = 1$

Now combine these results to find $f(-35, 5)$: $f(-35, 5) = -175 \cdot 1 = -175$

4. Verify and Summarize

We verified each component of the limit process, leading to the conclusion:

Final Answer

$\lim_{(x,y) \to (-35,5)} xy \cos(x + 7y) = -175$