Limits of Exponential,
Logarithmic, and Trigonometric
Function
GRADE 11 BASIC CALCULUS
Exponential and Logarithmic Functions
• If 𝑏 > 0, 𝑏 ≠ 0, the exponential function with base b is
defined by
𝒇 𝒙 = 𝒃𝒙 , 𝒙 ∈ ℝ
• Let 𝑏 > 0, 𝑏 ≠ 0. If 𝑏 𝑦 = 𝑥 then 𝑦 is called the logarithm
of 𝑥 the base b, denoted
𝒚 = 𝐥𝐨𝐠 𝒃 𝒙
Limits of Exponential Function
We will consider the natural exponential function
𝑓 𝑥 = 𝑒 𝑥 , where 𝑒 is called the Euler number, and
has value of 2.718281…
Example: Evaluate the lim 𝑒
𝑥→0
𝑥
Limits of Logarithmic Function
We will consider the natural logarithmic function
𝑓 𝑥 = ln 𝑥 . Recall that ln 𝑥 = log 𝑒 𝑥 . Moreover, it is
the inverse of the natural exponential function 𝑦 =
𝑒𝑥.
Example: Evaluate the lim ln 𝑥
𝑥→1
Limits of Logarithmic Function
Evaluate lim log 𝑥
𝑥→1
Limits of Trigonometric Functions
Evaluate lim sin 𝑥
𝑥→0
Example
𝑥
Evaluate lim 3
𝑥→1
Example
𝑥
Evaluate lim 5
𝑥→2
Example
Evaluate lim log 𝑥
𝑥→4
Example
Evaluate lim cos 𝑥
𝑥→0
Example
Evaluate lim tan 𝑥
𝑥→0
Example
Evaluate lim cos 𝑥
𝑥→𝜋
Example
Evaluate lim sin 𝑥
𝑥→𝜋
• lim cos 𝑥
𝑥→0
• lim cos 𝑥
𝑥→𝜋
• lim𝜋 cos 𝑥
𝑥→ 2