The thermal conductivity coefficient of a material is a function of temperature. What is the coefficient at the temperature of -∞, 1, +∞.
The function is:



Diagram of f(t) when t is from 0 to 5

When t = 1,

     f(t) = f(1)
          = 1 / [20 + (1 - 1)2]
          = 0.05

From the left diagram, it can be found that when t = 1, f(t) is equal to 0.05. The diagram verifies the previous calculated result.

Another method to solve this problem is to use the limit of the function.



Diagram of f(t)
when t is from -1000 to 0

When t = -∞,



Diagram of f(t)
when t is from 0 to 1000

When t = +∞,



  • Without limit concept
    In the case of t approaches -∞ and +∞, when the value of t is substituted into the function equation, the actual quantity of f(t) can not be calculated, because -∞ and +∞ are not actual number.
  • With limit concept
    As f(t) approaches -∞ ( +∞), for each positive ε, no matter how small, there is a corresponding positive δ, such that if