(ix)  If \(f(x) \le\ g(x),\ \forall\ \text{then}\ \lim\limits_{x \to a}\ g(x)\)

(x)  \(\lim\limits_{x \to a}[f(x)]^{g(x)}=\ \{\lim\limits_{x \to a}f(x)\}^{\lim\limits_{x \to a}g(x)}\)

(xi)  \(\lim\limits_{x \to a}f\{g(x)\}=\ f\{\lim\limits_{x \to a}\ g(x)\}=\ f(m)\) provided \(f\) is continuous at \(\lim\limits_{x \to a}\ g(x)=m\).

(xii)  Sandwich Theorem If \(f(x)\ \le g(x) \le\ h(x)\ \forall\ x\ \in(\alpha,\ \beta)-\ \{a\}\ \ and\ \ \lim\limits_{x \to a} f(x)=\ \lim\limits_{x \to a} h(x)=l,\) then, \(\lim\limits_{x \to a} g(x)=l,\ \text{where}\ a\ \in (\alpha,\ \beta)\)

Important Results on Limit

Some important results on limits are given below

1. Algebraic Limits

    (i)  \(\lim\limits_{x \to a}\ \frac{x^n-a^n}{x-a}=\ na^{n-1},\ n \in Q,\ a \gt 0\)

    (ii)  \(\lim\limits_{x \to \infty}\ \frac{1}{x^n}=0,\ n \in N \)

    (iii)  If \(m,\ n\) are positive integers and \(a_0,\ b_0\) are non-zero real numbers, then \(\lim\limits_{x \to \infty}\ \frac{a_0\ x^m\ +\ a_1\ x^{m-1}\ +\ ...\ +\ a_{m-1}\ x\ +\ a_m}{b_0\ x^n\ +\ b_1\ x^{n-1}\ +\ ...\ +\ b_{n-1}\ x\ +\ b_n}\)

      \(= \begin{cases} \frac{a_0}{b_0} & \quad \text{if }m=n\\ 0 & \quad \text{if } m \lt n\\ \infty & \quad \text{if } m \gt n, a_0 b_0 \gt 0\\ - \infty & \quad \text{if }m \gt n, a_0 b_0 \lt 0 \end{cases}\)

    (iv)  If \(\lim\limits_{x \to 0}\ \frac{(1\ +\ x)^n-1}{x}=n\)

    (v)  \(\lim\limits_{x \to 0}\ \frac{(1\ +\ x)^m-1}{(1\ +\ x)^n-1}=\frac{m}{n}\)

Trignometric Limits

(i)  \(\lim\limits_{x \to 0}\ \frac{\sin x}{x}=1= \lim\limits_{x \to 0}\ \frac{x}{\sin x}\)

(ii)  \(\lim\limits_{x \to 0}\ \frac{\tan x}{x}=1= \lim\limits_{x \to 0}\ \frac{x}{\tan x}\)

(iii)  \(\lim\limits_{x \to 0}\ \frac{\sin^{-1} x}{x}=1= \lim\limits_{x \to 0}\ \frac{x}{\sin^{-1} x}\)

(iv)  \(\lim\limits_{x \to 0}\ \frac{\tan^{-1} x}{x}=1= \lim\limits_{x \to 0}\ \frac{x}{\tan^{-1} x}\)

(v)  \(\lim\limits_{x \to 0}\ \frac{\sin x^\circ}{x}=\frac{\pi}{180}\)

(vi)  \(\lim\limits_{x \to \infty}\ \frac{\sin x}{x}= \lim\limits_{x \to \infty} \frac{\cos x}{x}=0\)

(vii)  \(\lim\limits_{x \to \infty} \sin\ x\ \ or\ \ \lim\limits_{x \to \infty} \cos\ x\) oscillates between \(-1\ to\ \ 1.\)

(viii)  \(\lim\limits_{x \to 0}\ \frac{\sin^P\ mx}{\sin^P\ nx}= \big(\frac{m}{n}\big)^P\)

(ix)  \(\lim\limits_{x \to 0}\ \frac{\tan^P\ mx}{\tan^P\ nx}= \big(\frac{m}{n}\big)^P\)

(x)  \(\lim\limits_{x \to 0}\ \frac{1\ -\ \cos m\ x}{1\ -\ \cos n\ x}=\frac{m^2}{n^2};\ \lim\limits_{x \to 0}\ \frac{\cos a x\ -\ \cos bx}{\ \cos c x\ -\ \cos\ dx}=\frac{a^2\ -\ b^2}{c^2\ -\ d^2}\)

(x)  \(\lim\limits_{x \to 0}\ \frac{\cos m\ x\ -\ \cos n\ x}{x^2}= \frac{n^2\ -\ m^2}{2}\)

 Logarithmic Limits

   (i)  \(\lim\limits_{x \to 0}\ \frac{\log_a(1\ +\ x)}{x}=\log_a\ e;\ a \gt 0,\ \ne 1\)

   (ii)  In particular, \(\lim\limits_{x \to 0}\ \frac{\log_e(1\ +\ x)}{x}=1\ \ and\ \ \lim\limits_{x \to 0}\ \frac{\log_e(1\ +\ x)}{x}=-1\)

Exponential Limits

   (i)  \(\lim\limits_{x \to 0} \frac{a^x\ -\ 1}{x}= \log_e\ a,\ a \gt 0\)

   (ii)  In particular, \(\lim\limits_{x \to 0} \frac{e^x\ -\ 1}{x}=1\ \ and\ \ \lim\limits_{x \to 0} \frac{e^{\lambda\ x}\ -\ 1}{x}=\lambda\)

  (iii)  \(\lim\limits_{x \to \infty}\ a^x = \begin{cases} 0, & \quad0 \le a\lt 1\\ 1, & \quad a=1\\ \infty, & \quad a \gt 1\\ \text{Does not exist, } & \quad a \lt 0 \end{cases}\)

5. \(1^\infty\ Form\ Limits\)

     \((i)\ \ \text{If}\ \lim\limits_{x \to a} f(x)= \lim\limits_{x \to a}\ g(x)=0, \text{then}\\ \ \ \ \ \ \ \lim\limits_{x \to a} \{1\ +\ f(x)\}^{1/g(x)}\ = e^{\lim\limits_{x \to a} \frac{f(x)}{g(x)}}\)

   \((ii)\ \ \text{If}\ \lim\limits_{x \to a} f(x)=1\ \ and\ \ \lim\limits_{x \to a}\ g(x)= \infty,\ \text{then}\\ \ \ \ \ \ \ \ \lim\limits_{x \to a} \{\ f(x)\}^{g(x)}\ = e^{\lim\limits_{x \to a} \{f(x)-1\}\ g(x)}\)

In General Cases

   (i)  \(\lim\limits_{x \to 0}\ (1\ +\ x)^{1/x}\ =e\)

   (ii)  \(\lim\limits_{x \to \infty}\ \Big(1+\ \frac{1}{x}\Big)^x=e\)

   (iii)  \(\lim\limits_{x \to 0}\ (1\ +\ \lambda\ x)^{1/x}\ =e^\lambda\)

   (iv)  \(\lim\limits_{x \to \infty}\ \Big(1+\ \frac{\lambda}{x}\Big)^x=e^\lambda\)

   (v)  \(\lim\limits_{x \to 0}\ (1\ +\ a\ x)^{b/x}\ = \lim\limits_{x \to \infty} \Big(1+\ \frac{a}{x}\Big)^{bx}=e^{ab}\)  

Methods to Evaluate Limits

To find \(\lim\limits_{x \to a}\ f(x),\) we substitute \(x=a\) in the function.

If \(f(a) \) is finite, then \(\lim\limits_{x \to a}\ f(x)= f(a).\)

If \(f(a)\) leads to one of the following form \(\frac{0}{0};\ \frac{\infty}{\infty};\ \infty\ -\ \infty;\ 0\ \times\ \infty;\ 1^\infty,\ 0\) and \(\infty^0\) (called indeterminate forms), then \(\lim\limits_{x \to a}\ f(x)\) can be evaluated by using following methods.

   (i)  Factorization Method  This method is particularly used when on substituting the value of \(x\), the expression take the form \(0/0.\)

   (ii)  Rationalization Method  This method is particularly used when either the numerator or the denominator or both involved square roots and on substituting the value of \(x\), the expression take the form \(\frac{0}{0};\ \frac{\infty}{\infty}.\)

\(\underline{\overline{NOTE}}\)

To evaluate \(x \to \infty\) type limits write the given expression in the form \(N/D\) and then divide both \(N\ \ and \ \ D\) by highest power of \(x\) occurring in both \(N\ \ and \ \ D\) to get a meaningful form.

L'Hospital's Rule

If \(f(x)\ \ and\ \ g(x)\) be two functions of \(x\) such that

   (i)  \(\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a}\ g(x) =0.\)

   (ii)  both are continuous at \(x=a\).

   (iii)  both are differentiable at \(x=a\).

   (iv)  \(f'(x)\ \ and\ \ g'(x)\) are continuous at the point \(x=a\), then \(\lim\limits_{x \to a} \frac{f(x)}{g(x)}= \lim\limits_{x \to a} \frac{f'(x)}{g'(x)}\) provided that \(g(a) \ne 0.\)

Above rule is also applicable, if \(\lim\limits_{x \to a} f(x)= \infty\ \ and\ \ \lim\limits_{x \to a}\ g(x)= \infty.\)

If \(f'(x),\ g'(x)\) satisfy all the conditions embedded in L' Hospital's rule, then we can repeat the application of this rule on

\( \frac{f'(x)}{g'(x)}\ \ \text{to get}\ \lim\limits_{x \to a}\ \frac{f'(x)}{g'(x)}= \lim\limits_{x \to a}\ \frac{f''(x)}{g''(x)}.\)

Sometimes, following expansions are useful in evaluating limits.

\(\bullet\ \ \log(1\ +\ x)=x\ -\ \frac{x^2}{2}\ +\ \frac{x^3}{3}\ +\ \frac{x^4}{4}\ +\ \frac{x^5}{5}\ + ...\ +\ (-1 \lt x \le 1)\)

\(\bullet\ \ \log(1\ -\ x)=\ -x\ -\ \frac{x^2}{2}\ -\ \frac{x^3}{3}\ -\ \frac{x^4}{4}\ -\ \frac{x^5}{5}\ - ...\ (-1 \lt x \lt 1)\)

\(\bullet\ \ e^x=1\ +\ \frac{x}{1!}\ +\ \frac{x}{2!}\ +\ \frac{x}{3!}\ +\ \frac{x}{4!}\ +\ ...\)

\(\bullet\ \ a^x=1\ +\ x\ (\log_e\ a)\ +\ \frac{x^2}{2!}\ (\log_e\ a)^2\ +...\)

\(\bullet\ \ \sin\ x=x\ -\ \frac{x^3}{3!}\ +\ \frac{x^5}{5!}\ -\ \frac{x^7}{7!}\ +\ ...\)

\(\bullet\ \ \cos\ x=1\ -\ \frac{x^2}{2!}\ +\ \frac{x^4}{4!}\ -\ \frac{x^6}{6!}\ +\ ...\)

Limits

Let \(y=f(x) \) be a function of \(x\). If the value of \(f(x)\) tend to a definite number as \(x\) tends to \(a\), then the number so obtained is called the limit of \(f(x) \) at \(x=a\) and we write it as \(\lim\limits_{x \to a}\ f(x).\)

\(\bullet\)  If \(f(x)\) approaches to \(l_1\)as \(x\) approaches to \('a'\) from left, then \(l_1\) is called the left hand limit of \(f(x)\) at \(x=a\) and symbolically we write it as \(f(a-0)\) or \(\lim\limits_{x \to a^-}\ f(x)\) or \(\lim\limits_{h \to 0}\ f(a-h)\)

\(\bullet\)  Similarly, right hand limit can be expressed as \(f(a+0)\ \ or\ \ \lim\limits_{x \to a^-}\ f(x)\ \ or\ \ \lim\limits_{h \to 0}\ f(a+h)\)

\(\bullet\)  \(\lim\limits_{x \to a^-}\ f(x)\) exists iff \(\lim\limits_{x \to a^-}\ f(x)\ \ and\ \ \lim\limits_{x \to a^+}\ f(x)\) exists and equal.

If \(\lim\limits_{x \to a} f(x)=l\ \ and\ \ \lim\limits_{x \to a}g(x)=m\) (where, \(l\ \ and\ \ m\) are real numbers), then

(i)  \(\lim\limits_{x \to a}\ \{f(x)\ +\ g(x)\}=l\ +\ m\)                    [sum rule]

(ii)  \(\lim\limits_{x \to a}\ \{f(x)\ -\ g(x)\}=l\ -\ m\)          [difference rule]

(iii)  \(\lim\limits_{x \to a}\ \{f(x)\ \cdot\ g(x)\}=l\ \cdot\ m\)                 [product rule]

(iv)  \(\lim\limits_{x \to a}\ k \cdot f(x)=k \cdot\ l\)              [constant multiple rule]

(v)  \(\lim\limits_{x \to a}\ \frac{f(x)}{g(x)}= \frac{l}{m},\ m\ne0\)                         [quotient rule]

(vi)  If \(\lim\limits_{x \to a} f(x)=+\ \infty,\ or\ \ -\ \infty,\text{then}\ \ \lim\limits_{x \to a} \frac{1}{f(x)}=0\)

(vii)  \(\lim\limits_{x \to a}|f(x)|=\lim\limits_{x \to a}|f(x)|\)

(viii)  \(\lim\limits_{x \to a}\ \log\{f(x)\}= \log\{\lim\limits_{x \to a}\ f(x)\},\) provided \(\lim\limits_{x \to a} f(x) \gt 0\)