Find the limit of (x2 - y2)/(x2 - y2) as (x,y) approaches (0,0) along line : (a)

Question

Find the limit of (x2 - y2)/(x2 - y2) as (x,y) approaches (0,0) along line : (a) y=x, (b) y=mx+b This is the problem from textbook (Advanced Calc 3rd Edition by Taylor & Mann). I'm not sure what it means to find limit along a line or how to begin the problem. Can you please walk me through the steps atleast for (a) and explain the difference between approaches for the different lines. Thanks.
Find the limit of (x2 - y2)/(x2 - y2) as (x,y) approaches (0,0) along line : (a) y=x, (b) y=mx+b This is the problem from textbook (Advanced Calc 3rd Edition by Taylor & Mann). I'm not sure what it means to find limit along a line or how to begin the problem. Can you please walk me through the steps atleast for (a) and explain the difference between approaches for the different lines. Thanks.

Explanation / Answer

When you say find the limit as you approach a point(in this case (0,0)) along the line, say y=mx+b, then what you mean is that you may assume that y = mx ; that is what it translates into, in terms of calculations. It should now be a routine matter to translate this into calculating the limit. Note that if you are approaching (0,0) along the line y = mx + b, then you must have b = 0 else, you will not approach the point (0,0). In more general terms, when you say approaching a point in 2 dimensions, it is really from from unambiguous, what it means. On the real line, there is only one sense of motion, so to speak. But in 2d, one needs to be a little more clear. To say that a limit exists in 2d, then it means NO MATTER WHAT the trajectory, the resulting calculation will give the same answer.

Related Questions

Navigate

• Browse (All)
• Browse F
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.