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Finding a Limit Using Spherical Coordinates In Exercises 77 and 78, use spherical coordinates to find the limit.[ Hint: Let
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Calculus: Early Transcendental Functions
- Tutorial Exercise Use polar coordinates to find the limit. [Hint: Let x = r cos 0 and y = r sin 0, and note that (x, y) → (0, 0) implies r → 0.] x2 - y2 lim (x, y) → (0, 0) x² + y² Step 1 Substitute the polar conversion factors x = r cos 0 and y = r sin 0 into the expressions x + y2 and x2 - y. x² + y? p2 cos² 0 + r² sin? 0 = 2(cos? 0 + %D x² – y2 = r² cos² e – r² sin² e p² (cos? 0 –arrow_forward(a) Show that the limit does not exist by considering the limits as (x, y) →(0,0) along the coordinate axes. x+y lim (x,y) 700) 2xy (b) Evaluate the limit by converting to polar coordinates. lim (x)-(0,0) In (x² + y²)arrow_forward= arctan(2), Find f, and f, and evaluate each at the given point. f(x, y) = arctan (4, -4) f,(x, y) = f,(x, y) = f(4, -4) = | f,(4, -4) = [arrow_forward
- Limitsa. Find the limit: lim (x,y)→(1,1) (xy - y - 2x + 2) / (x - 1).b. Show that lim (x,y)→(0,0) ((x - y)^2) / (x^2 + xy + y^2) does not existarrow_forwardUsing Green's theorem find the value of f F.drWhere F(x,y) = (e* – y³)i + (cosy + x³)j and C is the closed triangle bounded by the lines x = 0, y = 0 and x + y = 2.arrow_forwardUse polar coordinates to find the limit. [If (r, ?) are polar coordinates of the point (x, y) with r ≥ 0, note that r → 0+ as (x, y) → (0, 0).] (If an answer does not exist, enter DNE.) lim (x, y)→(0, 0) (x2 + y2) ln(x2 + y2)arrow_forward
- Determine whether the limit exists. If so, find its value. If the limit does not exist, indicate that using the check box. sin(x² + y? + z²) lim (x,y,z)→(0,0,0) Does not exist Vx2 + y? + z²arrow_forwardUse polar coordinates to find the limit. [If (r, 0) are polar coordinates of the point (x, y) with r2 0, note that r- 0+ as (x, y) → (0, 0).] (If an answer does not exist, enter DNE.) 4e-x2 - y? - 4 x² + y2 lim (x, y)- (0, 0)arrow_forwardsecy-1 Lim sec2y-1 a) as y → 0 b) as y → n/2arrow_forward
- If it exists, find Answer: x³ - ze²y lim Write DNE, if the limit does not exist. [Show solution.] (x,y,z) (-1,0,4) 6x + 2y - 3zarrow_forward* Let (X, d) and (Y, e) be metric spaces. Show that the function f: X xY X defined by f((x, y)) = x is continuous.arrow_forwardDIigisayar M Which of the following functions have limit ? 4xy 4 (x,y)- (0,0) x" +y lim = ? b) lim sin( xy) lim (x,y )- (0,0) x+y (x,y ) - ( 0,1) X +y-1 O a. None of them O b. all of them O c.a O d. b O e. a and b a Sonraki sayfa O AI PC TJ ak için buraya yazınarrow_forward
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