Limits

Limits at Infinity – Definition and Examples

Here you will learn how to solve or evaluate limits at infinity with examples. Let’s begin – Limits at Infinity Algorithm to evaluate limits at infinity : 1). Write down the given expression in the form of a rational function i.e. $$f(x)\over g(x)$$, if it is not so. 2). If k is the highest power …

Rationalisation Method to Solve Limits

Here you will learn what is the rationalisation method to solve or find limits with examples. Let’s begin – Rationalisation Method to Solve Limits This method is particularly used when either the numerator or denominator or both involve expression consisting of square roots and substituting the value of x the rational expression takes the form …

Factorisation Method to Solve Limits

Here you will learn what is the factorisation method to solve limits with examples. Let’s begin – Factorisation Method to Solve Limits Consider the following limit : $$\displaystyle{\lim_{x \to a}}$$ $$f(x)\over g(x)$$ If by substituting x = a, $$f(x)\over g(x)$$, reduces to the form $$0\over 0$$, then (x – a) is a factor of f(x) …

Direct Substitution Method to Solve Limits

Here you will learn direct substitution method to solve limits with examples. Let’s begin – Direct Substitution Method to Solve Limits Consider the following limits : (i)  $$lim_{x \to a}$$ f(x) (ii)  $$lim_{x \to a} {\phi(x)\over \psi(x)}$$ If f(a) and $$\phi(a)\over \psi(a)$$ exist and are fixed real numbers, then we say that $$lim_{x \to a}$$ …

Limits Examples

Here you will learn some limits examples for better understanding of limit concepts. Example 1 : If $$\displaystyle{\lim_{x \to \infty}}$$($${x^3+1\over x^2+1}-(ax+b)$$) = 2, then find the value of a and b. Solution : $$\displaystyle{\lim_{x \to \infty}}$$($${x^3+1\over x^2+1}-(ax+b)$$) = 2 $$\implies$$ $$\displaystyle{\lim_{x \to \infty}}$$$$x^3(1-a)-bx^2-ax+(1-b)\over x^2+1$$ = 2 $$\implies$$ $$\displaystyle{\lim_{x \to \infty}}$$$$x(1-a)-b-{a\over x}+{(1-b)\over x^2}\over 1+{1\over x^2}$$ = …

What is Squeeze Theorem – Limit of Exponential Functions

Here, you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples. Let’s begin – Squeeze Theorem (Sandwich Theorem) If f(x) $$\leq$$ g(x) $$\leq$$ h(x); $$\forall$$ x in the neighbourhood at x = a and $$\displaystyle{\lim_{x \to a}}$$ f(x) = l = $$\displaystyle{\lim_{x \to 1}}$$ h(x) then …

How to Find Limit of Trigonometric Functions

Here, you will learn how to find limit of trigonometric functions and limits using series expansion with example. Let’s begin –  Limit of Trigonometric Functions $$\displaystyle{\lim_{x \to 0}}$$ $$sinx\over x$$ = 1 = $$\displaystyle{\lim_{x \to 0}}$$ $$tanx\over x$$ = $$\displaystyle{\lim_{x \to 0}}$$ $$tan^{-1}x\over x$$ = $$\displaystyle{\lim_{x \to 0}}$$ $$sin^{-1}x\over x$$ [where x is measured in …

How to Solve Indeterminate Forms of Limits

Here, you will learn how to solve indeterminate forms of limits and general methods to be used to evaluate limits with examples. Let’s begin –  Indeterminate Forms of Limits $$0\over 0$$, $$\infty \over \infty$$, $$\infty – \infty$$,$$0\times \infty$$, $$1^{\infty}$$, $$0^0$$, $${\infty}^0$$ Note : (i)  Here 0, 1 are not exact, infact both are approaching to …

Definition of Limit in Calculus – Theorem of Limit

Here, you will learn definition of limit in calculus, left hand limit, right hand limit and fundamental theorem of limit. Let’s begin – Definition of Limit in Calculus Let f(x) be defined on an open interval about ‘a’ except possibly at ‘a’ itself. If f(x) gets arbitrarily close to L(a finite number) for all x …