Find \lim_{(x,y) \rightarrow (2,-3)} \frac{-5x^2-2y^2-1}{x^2+y^2-5}


Find {eq}\displaystyle \lim_{(x,y) \rightarrow (2,-3)} \frac{-5x^2-2y^2-1}{x^2+y^2-5} {/eq}

Limits by Direct Substitution:

Suppose we want to find the value of the limit {eq}\displaystyle\lim_{(x,y) \rightarrow (a,b) } f(x,y) {/eq}.

By direct substitution, we have:

{eq}\displaystyle\lim_{(x,y) \rightarrow (a,b) } f(x,y)=f(a,b) {/eq}

Note that we only apply direct substitution if the function {eq}f(x,y) {/eq} is continuous at {eq}(x,y) = (a,b) {/eq}.

Answer and Explanation:

The function {eq}\displaystyle \frac{-5x^2-2y^2-1}{x^2+y^2-5} {/eq} has continuity at {eq}(x,y) =(2,-3) {/eq} so we directly substitute...

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Learn more about this topic:

How to Determine the Limits of Functions

from Math 104: Calculus

Chapter 6 / Lesson 4

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