Limits

Question 1

lim_x→a k•ƒ(x)

Answer 1

k•lim_x→a ƒ(x)

Question 2

If lim_x→a ƒ(x) = L₁ and

lim_x→a g(x) = L₂, then

(Addition)

Answer 2

lim_x→a[ƒ(x) + g(x)] = L₁ + L₂

Question 3

If lim_x→a ƒ(x) = L₁ and

lim_x→a g(x) = L₂, then

(Multiplication)

Answer 3

lim_x→a [ƒ(x) • g(x)] = L₁ • L₂

Question 4

If the highest power of x in a rational expression is in the numerator, then the limit as x approaches infinity is?

Answer 4

Infinity

Example: lim_x→∞(5x⁷-3x)/(16x⁶-3x²) = ∞

Question 5

If the highest power of x in a rational expression is in the denominator, then the limit as x approaches infintiy is?

Answer 5

zero

Example: lim_x→∞ (5x⁶-3x)/(16x⁷-3x²) = 0

Question 6

If the highest power of x in a rational expression is the same in both the numerator and denominator, then the limit as x approaches infinity is?

Answer 6

The coefficient of the highest term in the numerator divided by the coefficient of the highest term the denominator.

Example lim_x→∞ (5x⁷-3x)/(16x⁷-3x²) = 5/16

Question 7

lim_x→∞ c/x

Answer 7

= 0

Question 8

If |x|>1 and n>0, then

lim_x→∞ k/xⁿ = ?

lim_x→-∞ k/xⁿ = ?

Answer 8

= 0

= 0

Question 9

lim_x→±∞ (x/x)

Answer 9

1

Question 10

lim_x→0 sinx/x

(x is in radians)

Answer 10

1

Question 11

lim_x→0 (cosx - 1)/x

Answer 11

0

Question 12

lim_x→0 sin(ax)/x

Answer 12

a

Question 13

lim_x→0 sin(ax) / sin(bx)

Answer 13

a/b

Question 14

L’Hopital’s Rule

Answer 14

If ƒ(x) = g(x)/h(x) and if

lim_x→cg(x) = lim_x→ch(x) = 0 or ∞

then lim_x→cƒ(x) = lim_x→c g‘(x)/h‘(x)