Evaluation of Limits
There are various methods for evaluation of limits such as:
Substitution: This is the simplest method, where we just plug in the value of the limit into the function and see if it works. For example,
lim x → 2 x2 + 3x − 4 = (2)2 + 3(2) – 4 = 4 + 6 – 4 = 6.
Factoring: This method is helpful when substitution gives us an indeterminate form, such as 0/0 or 10, where k is a constant. In this case, we can try to factor the numerator and denominator of the function and cancel out any common factors. For example, limx → 2 (x2 – 4)/(x − 2). If we substitute x = 2 here, limit becomes 0/0 form. Thus, it can be calculated as:
limx → 2 (x2 – 4)/(x − 2) = limx → 2 [(x – 2)(x + 2)]/(x − 2) = limx → 2 (x + 2) = 2 + 2 = 4
Algebraic manipulation: This method involves simplifying the function by using algebraic rules, such as expanding, combining, or dividing terms. For example,
limx → 0 (x2 + 2x)/(x2 – 2x) = limx → 0 x(x + 2)/x(x – 2) = limx → 0 (x + 2)/(x – 2) = (0 + 2)/(0 – 2) = 2/(-2) = -1.
L Hospital Rule
If limx → a f(x)/g(x) is in the form of 0/0 or ∞/∞ then
[Tex]\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}[/Tex]
Where f'(x) and g'(x) are the first order derivatives of functions f(x) and g(x) respectively.
Other Indeterminant forms in limits are:
- 0 ⋅ ∞
- ∞ − ∞
- 00
- 1∞
- ∞0
Read More about L Hospital Rule.
Sandwich Theorem
For functions f(x), g(x), and h(x) such that f(x) ≤ g(x) ≤ h(x) then for some value ‘a’ if [Tex] \bold{\lim_{x \to a}f(x)} = p = \bold{\lim_{x \to a}h(x)} [/Tex]then
[Tex]\bold{\lim_{x \to a}g(x)= p}[/Tex]
Read more about Sandwich Theorem.
Calculus Cheat Sheet
Calculus is a branch of mathematics that studies the properties and behavior of functions, rates of change, limits, and infinite series. Calculus has many applications in science, engineering, economics, and other fields. However, calculus can also be challenging to learn and master, especially for beginners.
That is why we have prepared this calculus cheat sheet, a handy reference guide covering the most important concepts, formulas, rules, and calculus examples. Whether you need a quick review, a study aid, or a problem solver, this cheat sheet will help you ace calculus with ease.
Read: Calculus in Maths