Evaluation of Limits
Limits can be solved with different methods depending on the type of form it exhibits for x = a.
- Determinate Forms
- Indeterminate Forms
Determinate Forms
If at x = a, f(x) yields a definite value then the limit is calculated by \lim_{x \to a}f(x)=f(a).
Indeterminate Forms
If at x = a, f(x) yields a value in the form of 0/0, ∞/∞, ∞-∞, 00,1∞, and ∞0 then they are called Indeterminate Forms. It can be solved by following mentioned methods:
- Factorization Method
It is used when [Tex]\lim_{x \to a}\frac{f(x)}{g(x)} [/Tex]takes the form of 0/0 then x-a is a factor of the numerator and denominator which can be cancelled to make it into determinate form and then solve.
- Rationalization Method
This method is used when [Tex]\lim_{x \to a}\frac{f(x)}{g(x)} [/Tex]takes the form of 0/0 or ∞/∞ and the denominator is in square root form. In this case, the denominator is rationalized.
- Substitution Method
In this case, the x in f(x) is replaced with x = a + h or a – h such that when x tends to a then h tends to 0.
When x→∞: In this case when [Tex]\lim_{x \to ∞}\frac{f(x)}{g(x)} [/Tex]takes the form of ∞/∞ then the numerator and denominator are divided by the highest power of x.
Learn More, Strategy in Finding Limits
L Hospital Rule
L Hospital Rule states that if f(x)/g(x) is in the form of 0/0 or ∞/∞ for x = a then \lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)} , where f'(x) and g'(x) are the first order derivatives of functions f(x) and g(x) respectively.
Sandwich Theorem
Sandwich Theorem states that for given functions f(x), g(x), and h(x) that exists in the order f(x) ≤ g(x) ≤ h(x) for x belonging to the common domain then for some value ‘a’ if [Tex] \bold{\lim_{x \to a}f(x)} = p = \bold{\lim_{x \to a}h(x)} [/Tex] then \bold{\lim_{x \to a}g(x)} = p
Differential Calculus
Differential Calculus is a branch of Calculus in mathematics that is used to find rate of change of a quantity with respect to other. It involves calculating derivatives and using them to solve problems involving non constant rates of change. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications.
In this article, we have tried to provide a brief overview of the branch of Differential Calculus including topics such as limits, derivatives, various formulas for derivatives as well as application of derivatives.
Table of Content
- What is Differential Calculus?
- What is Limit?
- Limit Formulas
- Continuity, Discontinuity, and Differentiability of a Function
- Derivatives
- Differentiation Formulas
- Implicit Differentiation
- Higher Order Derivatives
- Error
- Approximation
- Inflection Point
- Tangent and Normal
- Increasing and Decreasing Function
- Maxima and Minima
- Extreme Value Theorem
- First Derivative Test
- Second Derivative Test
- Differential Equation