How do you find the limit as (x,y) approaches (0,0) of #(x+y^2) / (2x+y)#?
You cannot do There is no way you can rewrite this without it be undefined. This is the graph:
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To find the limit as (x,y) approaches (0,0) of (x+y^2) / (2x+y), we can use the limit definition. First, we substitute the values of x and y into the expression:
lim(x,y)→(0,0) (x+y^2) / (2x+y)
Next, we can try approaching the limit along different paths. Let's consider approaching along the x-axis (y=0):
lim(x,y)→(0,0) (x+0^2) / (2x+0) lim(x,y)→(0,0) x / (2x) lim(x,y)→(0,0) 1/2
Now, let's approach along the y-axis (x=0):
lim(x,y)→(0,0) (0+y^2) / (0+y) lim(x,y)→(0,0) y^2 / y lim(x,y)→(0,0) y
Since the limit approaches different values along different paths, the limit does not exist as (x,y) approaches (0,0) for the given expression.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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