Show that lim (x, y)→(0, 0) 2x2 − 5y2 x2 + y2 does not exist. SOLUTION Let f(x, y) = (2x2 − 5y2)/(x2 + y2). First, let's approach (0, 0) along the x-axis. Then y = gives f(x, 0) = 2x2/x2 = for all x ≠ 0, so f(x, y) → as (x, y) → (0, 0) along the x-axis. We now approach along the y-axis by putting x = . Then f(0, y) = −5y2/y2 = for all y ≠ 0, so f(x, y) → as (x, y) → (0, 0) along the y-axis. Since f has two different limits along two different lines, the given limit does not

In a column proof the right column states your reasons

In δabc shown below, ∠bac is congruent to ∠bca: triangle abc, where angles a and c are congruent given: base ∠bac and ∠acb are congruent. prove: δabc is an isosceles triangle. when completed (fill in the blanks), the following paragraph proves that line segment ab is congruent to line segment bc making δabc an isosceles triangle. (4 points) construct a perpendicular bisector from point b to line segment ac . label the point of intersection between this perpendicular bisector and line segment ac as point d: m∠bda and m∠bdc is 90° by the definition of a perpendicular bisector. ∠bda is congruent to ∠bdc by the definition of congruent angles. line segment ad is congruent to line segment dc by by the definition of a perpendicular bisector. δbad is congruent to δbcd by the line segment ab is congruent to line segment bc because consequently, δabc is isosceles by definition of an isosceles triangle. 1. corresponding parts of congruent triangles are congruent (cpctc) 2. the definition of a perpendicular bisector 1. the definition of a perpendicular bisector 2. the definition of congruent angles 1. the definition of congruent angles 2. the definition of a perpendicular bisector 1. angle-side-angle (asa) postulate 2. corresponding parts of congruent triangles are congruent (cpctc)

1. according to the internal revenue service, the mean tax refund for the year 2007 was $2,708. assume the standard deviation is $650 and that the amounts refunded follow a normal probability distribution. a. what percent of the refunds are more than $3,000? b. what percent of the refunds are more than $3,000 but less than $4,000? c. what percent of the refunds are less than $2,000?