Indeterminate Forms

In-determinate Forms

If f(x) at x = a assumes forms like etc., which do not represent any value are called In-determinate forms. The concept of limit gives a meaningful value for the function  at overcomes these in-determinate forms. The differentiations for such forms are performed using L’Hospital’s (French Mathematician) rule.

L’ Hospital’s theorem

Statement

If  f(x) and g(x) are two functions such that

Problems are categorized under three different types.

  • Type 1
  • Type 2
  • Type 3

Note

Standard limits that are used in the problems are

Type 1 Problems

  1. Evaluate lim (x→0) (x e^x -log(1+x)) / x^2
  2. Evaluate lim (x→π/2) log(sin x) / (π – 2x)^2
  3. Evaluate lim (x→0) (sinh x – x) / ( sin x – x cos x)
  4. Evaluate lim (x→π/2) log(x-π/2) / tan x
  5. Evaluate lim (x→0) [a/x – cot x/a]
  6. Evaluate lim (x→0) [1/x – log(1+x)/x^2]

Type 2 Problems

  1. Evaluate lim x -> 0 (tan x – x)/(x ^ 2 * tan x)
  2. Evaluate lim (x->0) (x ^ 2 + 2 cos x – 2 ) / ( x sin^3 x)
  3. Evaluate lim (x->0) ((1/x^2)-(1/sin^2 x))
  4. Evaluate lim (x->0) ((1/x^2)-(cot^2 x))
  5. Evaluate lim (x->0) ((e^x – e^(-x) – 2 log(1+x)) / (x sinx) )
  6. Evaluate lim (x->0) ( (1 + sinx – cosx + log(1-x)) / (x tan^2 x) )

Type 3 Problems

  1. Evaluate lim(x→π/2)(sinx)^(tanx)
  2. Evaluate lim(x→0) (tanx/x)^(1/x)
  3. Evaluate lim(x→0) ((a^x+b^x)/2)^(1/x)
  4. Evaluate lim(x→0) ((a^x+b^x+c^x)/3)^(1/x)
  5. Evaluate lim(x→0) ((a^x+b^x+c^x+d^x)/4)^(1/x)
  6. Evaluate lim(x→0) [(sin^2(π/(2-x))]^sec^2(π/(2-x))

Related Topics

 

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