# Math 254 § 14 The definition of the limit_ tangent plane and the by huangyuarong

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Math 254                                                       TA: Gangyong Lee

§ 14 The deﬁnition of the limit, tangent plane and the partial deriva-
tive

1) In the function y = f (x) if the value of the right limit and the value
of the left limit are the same then there exists the limit whose value is
the same as the above value. That is,

if lim− f (x) = lim+ f (x) then lim f (x) = lim− f (x) = lim+ f (x)
x→a          x→a              x→a             x→a            x→a

We call lim f (x) = A
x→a
if for any , there exists δ > 0 such that |x − a| < δ ⇒ |f (x) − A| < .

2) In the function z = f (x, y) we call      lim        f (x, y) = A
(x,y)→(a,b)
if for any , there exists δ > 0 such that          (x − a)2 + (y − b)2 < δ ⇒
|f (x, y) − A| < .

3) tangent line
i) the general linear equation; ax + by + c = 0
ii) the equation of the line which goes through the point (x0 , y0 );
a(x − x0 ) + b(y − y0 ) = 0
iii) the equation of the tangent line of y = f (x) at the point (x0 , y0 );
y − y0 = f (x0 )(x − x0 )

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4) tangent plane
i) the general plane equation; ax + by + cz + d = 0
ii) the equation of the plane which goes through the point (x0 , y0 , z0 );
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0
iii) the equation of the tangent plane of z = f (x, y) at the point
(x0 , y0 , z0 ); z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 )

5) The diﬀerence between the derivative and the partial derivative.
i) If we represent z as the function only w.r.t. t then we use dz .
dt
ii) If we represent z as the function with respect to more than 1 variable
then we use ∂z .
∂t

y
Example #5 In the problem w = xe z and x = t2 , y = 1−t, z = 1+2t
1−t
since w = t2 e 1+2t we have to ﬁnd dz instead of ∂z .
dt            ∂t

Example #7 In the problem z = x2 + xy + y 2 and x = s + t, y = st
since z = (s + t)2 + (s + t)st + (st)2 we have to ﬁnd ∂z and ∂z instead
∂s     ∂t
of dz and dz , respectively.
ds     dt

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