If the limit of a function does not exist, it is written as lim 𝑥→𝑐 𝑓(𝑥) 𝐷𝑁𝐸 but NOT lim 𝑥→𝑐 𝑓(𝑥) = 𝐷𝑁𝐸 because 𝐷𝑁𝐸 is not a value. It just indicates that the functions move in different directions as its variable approaches 𝑐 from the left and from the right.

For a limit to exist, the limit from the left and from the right must both
exist and be equal to L.

3.2

In the radical or root theorem, if 𝑛 is even, the value of the limit 𝐿 must be positive; otherwise, if it is negative, the limit does not exist.

If 𝑓 is any function other than the polynomial function, then lim 𝑥→𝑐 𝑓(𝑥) is not always equal to 𝑓(𝑐).

3.2a

A limit that is in determinate form of type may exist using factoring or by
rationalizing.

3.3

Remember that ∞ is not a number, lim 𝑥→c 𝑓 𝑥 = +∞ and lim 𝑥→𝑐 𝑓(𝑥) = −∞ describes the behavior of the function near 𝑐.

To evaluate lim 𝑥→+∞ 𝑓(𝑥)/𝑔(𝑥) or lim 𝑥→−∞ 𝑓(𝑥)/𝑔(𝑥), where 𝑓(𝑥) and 𝑔(𝑥) are polynomials, divide both the numerator and denominator by the highest power of 𝑥 that occurs in the denominator.

3.3a

+∞/k = +∞, k>0 (k is positive)

+∞/k = -∞, k<0 (k is negative)

-∞/k = -∞, k>0 (k is positive)

-∞/k = +∞, k<0 (k is negative)

To evaluate the limits at infinity of polynomials, factor out the highest exponent.

If lim 𝑥→𝑐+ 𝑓(𝑥) or lim 𝑥→𝑐− 𝑓(𝑥) equals to +∞ or −∞, the line 𝑥 = 𝑐 is called a vertical asymptote for 𝑓(𝑥).

If lim 𝑥→+∞ 𝑓(𝑥) or lim 𝑥→−∞ 𝑓(𝑥) equals to 𝐿, the line y = 𝐿 is called a horizontal asymptote for 𝑓(𝑥).

(no idea what those 2 meant)

3.4

If the given is the graph of a function, it is continuous on the interval if the graph can be traced out without lifting our hands.

3.4a

It is said to be removable because the discontinuity may be removed by
redefining 𝑓(𝑐) that it will be equal to the lim 𝑥→𝑐 𝑓(𝑥) = 𝐿.

If lim 𝑥→𝑐 𝑓(𝑥) = 𝐿, a removable discontinuity is remedied by the redefinition: 𝑓 (𝑐) = 𝐿.

The value of 𝑐 ∈ [𝑎, 𝑏] in the conclusion of the Intermediate Value Theorem is not necessarily unique.

Similar to the IVT, the value 𝑐 ∈ [𝑎, 𝑏] at which a minimum or a maximum value occurs is not necessarily unique.

(no idea what those 2 meant, again)

3.5

The limit definition of the derivative is inherently indeterminate. Hence, the usual techniques for evaluating limits which are indeterminate of type 0/0 are applied.

1. If 𝑓 is continuous at 𝑥 = 𝑎, it does not mean that 𝑓 is differentiable at 𝑥 = 𝑎.
2. If 𝑓 is not continuous at 𝑥 = 𝑎, then 𝑓 is not differentiable at 𝑥 = 𝑎.
3. If 𝑓 is not differentiable at 𝑥 = 𝑎, it does not mean that 𝑓 is not continuous at 𝑥 = 𝑎.
4. A function 𝑓 is not differentiable at 𝑥 = 𝑎 if 𝑓 is not continuous at 𝑥 = 𝑎.

3.6

1. The sum can be extended to a sum of a finite number of functions. If 𝑓 𝑥 =
𝑓 1 𝑥 + 𝑓2 𝑥 + ⋯ + 𝑓𝑛 (𝑥) where 𝑓1 , 𝑓2 , … , 𝑓𝑛 are differentiable functions, then 𝑓 ′ 𝑥 = 𝑓1 ′ 𝑥 + 𝑓2 ′ 𝑥 + ⋯ + 𝑓𝑛 ′(𝑥).
2. It is also true for the difference of a finite number of functions. If 𝑓 𝑥 = 𝑓1 𝑥 − 𝑓2 𝑥 − ⋯ − 𝑓𝑛 (𝑥) where 𝑓1 , 𝑓2 , … , 𝑓𝑛 are differentiable functions, then 𝑓 ′ (𝑥) = 𝑓1 ′ 𝑥 − 𝑓2 ′(𝑥) − ⋯ − 𝑓𝑛 ′(𝑥).

D𝑥 [𝑓(𝑥)𝑔(𝑥) ≠ 𝐷𝑥[𝑓(𝑥) ⋅ 𝐷𝑥 [𝑔(𝑥)]

high/low = [(low)(high') - (high)(low)]/low^2

The derivative of a quotient is not equal to the quotient of their derivatives

3.7

1. The slope of the tangent is the derivative of a function at a given point
𝑃(𝑥0 , 𝑦0 ).

2. Two lines are said to be perpendicular if their slope is