Limit of Exponential,
Logarithmic, and
Trigonometric
Functions
1
𝑓 𝑥 = 𝑥−1 𝑥+1 𝑥+3
2
𝑥
𝑓 𝑥 =
𝑥−1
𝑓 𝑥 = 𝑥−1
𝑓 𝑥 = sin 𝑥
𝑓 𝑥 = cos 𝑥
𝑓 𝑥 = 𝑏𝑥
𝑔 𝑥 = log 2 𝑥
𝑓 𝑥 = tan 𝑥
𝑓 𝑥 = cot 𝑥
𝑓 𝑥 = sec 𝑥
𝑓 𝑥 = csc 𝑥
8
9
10
lim sin 𝑥 = sin 𝑎
𝑥→𝑎
lim cos 𝑥 = cos 𝑎
𝑥→𝑎
𝑥
lim 𝑏 = 𝑏
𝑥→𝑎
𝑎
11
If 𝑎 is in the domain of 𝑓(𝑥) and 𝑓 𝑥
is one of the following:
tan 𝑥 , cot 𝑥 , sec 𝑥 , csc 𝑥 , log 𝑏 𝑥 (𝑏 > 0),
then
lim 𝑓(𝑥) = 𝑓(𝑎)
𝑥→𝑎
Examples:
lim 𝑒
1. 𝑥→0
lim
3
2. 𝑥→1
3.
lim 5
𝑥→2
Exponential
𝑥
𝑥
𝑥
3𝑥
lim
4. 𝑥→0
4
4𝑥
−3
𝑥
𝑒 𝑥 − 𝑒 −𝑥
lim
5. 𝑥→0
𝑥
lim 𝑒
1. 𝑥→0
𝑥
lim
3
2. 𝑥→1
𝑥
3.
lim 5
𝑥→2
𝑥
3𝑥
lim
4. 𝑥→0
4
4𝑥
−3
𝑥
𝑒 𝑥 − 𝑒 −𝑥
lim
5. 𝑥→0
𝑥
𝑥
𝑒 −𝑒
lim
5. 𝑥→0
𝑥
−𝑥