CalculusI -
Intro to Limits
Goals of Calculus Ex : f(x) = x2
(x xy) slope
thearea
· (x y) m =
# findi S
T +
point
,
,
of ,
>
-
function
a
& As Q + P
P(1 1)
Mse
· >
Mean
-
,
Eg for tan : & X & I -> undefined
mcn(Xc )
> Site Y y = - X
, Q + P
points two
between
(
Man =
X -
I
X
I -
1
Tangent Problem
↳ becaus X already cannot and will not equl 1.
#decant
- X+ )
ms =
live What ?
happens to
msen as X-1
↳ More
Q mely close to P (but not 4) ↳
Msec-> 2
the secant live becomes (essentially) a
tangent line Because More-> 2
(slope of sea live approaches 2)
...
BUT ↳
Mean = 2 (slope of tan lime actually is .
2)
5For a line , you
need two points (P + Q) 1 = 2(x D -
y
-
↳ As Q approaches P ,
the secant
gets closer to the
y -
1 = 2x -
2
F2x- 11
>
tampant of P
Ein - >
& If Q -P but Q & P
,
the secant will be identical
t the tangent A limit must approach the same value from
↳ THIS A
IS "LIMIT" either side.
I General
limf(x) = L
*I
Spill in 4973 4 .
Sided Limit
Right
↳im f(x) ,
->
/im f(x) ,
↳im f(x) =
(imf(x)
No
For a limit to exist at a
↳ limf(x) =
limf(x)
ab
X> -
2 x -
, 2
Computing Limits
Lim (x-2x
Basics Lim
: c = c (constant
[imx] (in2) (in ** ·
Sim
Lim x
-a
23 -
2 .
207
*
Els ↓im (x2xx) =
= o
-
-
D
Xin = D
fla
↳im f(x)
=
Properties :
Lim f(x) =L
lim X- 4
#(x
2x2
(X -
2
2
2)
+
Xim g(x) L This isskay
x + 2 -
X
=
because crib
i
we
X bo = 2
1
m [f(x) g(x)] = imf(x) my
+
allowing
anyway
2 .
xina(f(x) g(x)] Ximf(x) /lim g(x)
· = ·
-
Lim
3
.
lim( x , Lim g(x lim
x+
3
x2 - 3x-10
x 25 -
- Limitdis
4 .
/ima [f(x)]" =
[limf(x)]" Note
↳ If 8 factor simplify
↳ Lim
and
= f(x) ↳ If you
,
can't simplify the Sinlysis
& I It
:
problem away
. .. It isn't a -
2
hole .
Evaluate
min
↳ sign analysis
>
-
N
Xim 1 1 2
= + =
+
lim M I ·
/
=
16X -
-
X- y X
lim
+ o ) = i
,
,
Intro to Limits
Goals of Calculus Ex : f(x) = x2
(x xy) slope
thearea
· (x y) m =
# findi S
T +
point
,
,
of ,
>
-
function
a
& As Q + P
P(1 1)
Mse
· >
Mean
-
,
Eg for tan : & X & I -> undefined
mcn(Xc )
> Site Y y = - X
, Q + P
points two
between
(
Man =
X -
I
X
I -
1
Tangent Problem
↳ becaus X already cannot and will not equl 1.
#decant
- X+ )
ms =
live What ?
happens to
msen as X-1
↳ More
Q mely close to P (but not 4) ↳
Msec-> 2
the secant live becomes (essentially) a
tangent line Because More-> 2
(slope of sea live approaches 2)
...
BUT ↳
Mean = 2 (slope of tan lime actually is .
2)
5For a line , you
need two points (P + Q) 1 = 2(x D -
y
-
↳ As Q approaches P ,
the secant
gets closer to the
y -
1 = 2x -
2
F2x- 11
>
tampant of P
Ein - >
& If Q -P but Q & P
,
the secant will be identical
t the tangent A limit must approach the same value from
↳ THIS A
IS "LIMIT" either side.
I General
limf(x) = L
*I
Spill in 4973 4 .
Sided Limit
Right
↳im f(x) ,
->
/im f(x) ,
↳im f(x) =
(imf(x)
No
For a limit to exist at a
↳ limf(x) =
limf(x)
ab
X> -
2 x -
, 2
Computing Limits
Lim (x-2x
Basics Lim
: c = c (constant
[imx] (in2) (in ** ·
Sim
Lim x
-a
23 -
2 .
207
*
Els ↓im (x2xx) =
= o
-
-
D
Xin = D
fla
↳im f(x)
=
Properties :
Lim f(x) =L
lim X- 4
#(x
2x2
(X -
2
2
2)
+
Xim g(x) L This isskay
x + 2 -
X
=
because crib
i
we
X bo = 2
1
m [f(x) g(x)] = imf(x) my
+
allowing
anyway
2 .
xina(f(x) g(x)] Ximf(x) /lim g(x)
· = ·
-
Lim
3
.
lim( x , Lim g(x lim
x+
3
x2 - 3x-10
x 25 -
- Limitdis
4 .
/ima [f(x)]" =
[limf(x)]" Note
↳ If 8 factor simplify
↳ Lim
and
= f(x) ↳ If you
,
can't simplify the Sinlysis
& I It
:
problem away
. .. It isn't a -
2
hole .
Evaluate
min
↳ sign analysis
>
-
N
Xim 1 1 2
= + =
+
lim M I ·
/
=
16X -
-
X- y X
lim
+ o ) = i
,
,
