Learning these TS Inter 1st Year Maths 1B Formulas Chapter 8 Limits and Continuity will help students to solve mathematical problems quickly.

## TS Inter 1st Year Maths 1B Limits and Continuity Formulas

→ If a variable x approaches a value a’ from the left i.e., through values just smaller than ’a’ than the limit of f defined is called the left limit of f(x) denoted by \(\lim _{x \rightarrow a^{-}}\)f(x)

\(\lim _{x \rightarrow a^{-}}\)f(x)= \(\lim _{h \rightarrow 0^{+}}\)f(a – h) = \(\lim _{x \rightarrow 0}\)f(a – x) (∵ x → a^{–} ⇒ x < a)

→ If x approaches a’ from the right i.e., through the values just greater than ‘a’ then the limit of f defined is called the right limit of f(x) denoted by \(\lim _{x \rightarrow a^{+}}\)(x).

\(\lim _{x \rightarrow a^{+}}\) f(x)= \(\lim _{h \rightarrow 0^{+}}\) f(a + h)= \(\lim _{x \rightarrow 0}\)f(a + x) (∵ x → a^{+} ⇒ x > a)

→ Suppose f is defined in a deleted neighbourhood of ‘a’ and l e R then

\(\lim _{x \rightarrow a}\)f(x) = l ⇒ \(\lim _{x \rightarrow a^{+}}\)f(x) = \(\lim _{x \rightarrow a^{-}}\)f(x) = l

→ Standard limits:

- \(\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}\) = na
^{n-1}and \(\lim _{x \rightarrow a}\left(\frac{x^m-a^m}{x^n-a^n}\right)=\frac{m}{n}\)a^{m-n} - \(\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)\) = 1, \(\lim _{x \rightarrow 0}\left(\frac{\tan x}{x}\right)\) = 1
- \(\lim _{x \rightarrow 0}\left(\frac{a^x-1}{x}\right)\) = log
_{e}^{a} - \(\lim _{x \rightarrow 0}\)(1 + x)
^{\(\frac{1}{x}\)}= e and \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^x\) = e - \(\lim _{x \rightarrow 0}\left(\frac{e^x-1}{x}\right)\) = 1

Note:

For finding \(\lim _{x \rightarrow a}\)f(x), first verify f(a). If this is in indeterminate form like \(\frac{0}{0}, \frac{\infty}{\infty}\) etc., then reduce the given limit into standard form or rationalise numerator or denominator or factorise according to the problem.