MATH 136
0 / 0 Indeterminate Limits
Give an algebraic derivation to find the exact value of the limit.
3
1
−
1. lim 93 x 2
x → 4 x − 64
3.
x3
+8
x → −2 2 − x + 6
lim
12x 2 − 100 x − 72
2. lim
x −3
x→ 9
10
−2
x
4. lim 2
x → 5 x − 25
Exercises
Solutions
1. Initially the limit is 0/0. So we algebraically manipulate the function to cancel terms
x − 4 (or 4 − x ) and eliminate the 0/0 indeterminate:
€
3
1
−
6 + 9x 
6 − 9x
36 − 9 x
9x 2 =

=
€3
2
x − 64
2 9x ( x − 4)(x + 4 x + 16)  6 + 9 x  2 9 x (x − 4)(x 2 + 4x + 16)( 6 + 9x )
=
−9( x − 4)
−9
=
, provided x ≠ 4.
2
2
2 9x ( x − 4)( x + 4x + 16)(6 + 9 x ) 2 9 x ( x + 4x + 16) (6 + 9x )
Re-evaluate:
3
1
−
−9
−9
−1
.
lim 93 x 2 = lim
=
=
2
768
x → 4 x − 64
x → 4 2 9 x (x + 4 x + 16)( 6 + 9x ) 6912
--------------------------------------------------------------------------------------------------------------------2. Because 9 is a root of the quadratic in the numerator, x − 9 is a factor; thus,
12 x 2 − 100x − 72 (x − 9)(12x + 8)  x + 3  (x − 9)(12 x + 8) ( x + 3 )
=

=
x −3
x −3
 x€ + 3 
x −9
= (12 x + 8)( x + 3 ), provided x ≠ 9.
12x 2 − 100 x − 72
= lim (12 x + 8)( x + 3 ) = 116 × 6 = 696.
x −3
x→ 9
x→9
--------------------------------------------------------------------------------------------------------------------Re-evaluate: lim
3. Because x = −2 is causing the 0/0 indeterminate, we try to eliminate x + 2 from the
numerator and denominator:
3
2

€x + 8 = ( x + 2)( x − 2x + 4)  2 +
2− x+6
2− x+6
2+
2
x + 4)( 2 +
x + 6  (x + 2)(x − 2€
=
x +6
4 − ( x + 6)
( x + 2)( x 2 − 2x + 4) ( 2 + x + 6 )
=
= −(x 2 − 2 x + 4)( 2 +
−x− 2
x + 6)
x + 6 ), provided x ≠ 2.
x3 + 8
= lim − ( x 2 − 2x + 4) ( 2 + x + 6 ) = −12 × 4 = −48 .
2
−
x
+
6
x → −2
x → −2
--------------------------------------------------------------------------------------------------------------------10
−2
10 − 2x
2(5 − x)
−2
x
=
=
=
, provided x ≠ 5
4. 2
x − 25 x(x − 5)( x + 5) x(x − 5)( x + 5) x( x + 5)
Re-evaluate:
lim
10
−2
−2
−2
1
= lim
=
=− .
Re-evaluate: lim x2
50
25
x → 5 x − 25 x → 5 x (x + 5)