Revision: 11th Std >> Limits MAH-MHT CET (PCM/PCB) Limits

When we have the values of f near x to the right of a i.e.
\[\lim_{x\to a^{+}}f\left(x\right)\] is the expected value of f at x = a.

If f(x) approaches a real number l, when x approaches a, then l is called the limit of f(x).

Symbolically, \[\lim_{x\to a}f\left(x\right)=l\]

When we have the values of f near x to the left of a, i.e.
\[\lim_{x\to a^{-}}f\left(x\right)\] is the expected value of f at x = a.

If\[\operatorname*{lim}_{x\to a}\frac{f(x)}{g(x)}\] is of the indeterminate form \[\frac{0}{0}\]​, then factorise f(x) and g(x) and cancel the common factors to evaluate the limit.

If we get \[\frac{0}{0}\] form and the numerator or denominator or both have a radical sign, then rationalise and substitute the limit.

If \[\lim_{x\to a}f\left(x\right)=l,\], where l is a real number, then l is the limit of the function.

1. \[\lim_{x\to0}\left(\frac{e^{x}-1}{x}\right)=\log e=1\]

2. \[\lim_{x\to0}\left(\frac{a^{x}-1}{x}\right)=\log a(a>0,a\neq1)\]

3. \[\lim_{x\to0}(1+x)^{\frac{1}{x}}=e=\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}\]

4. \[\lim_{x\to0}\left(\frac{\log\left(1+x\right)}{x}\right)=1\]

5. \[\lim_{x\to0}\left(\frac{e^{px}-1}{px}\right)=1,\] (p constant)

6. \[\lim_{x\to0}\left(\frac{a^{px}-1}{px}\right)=\log a,\] (p constant)

7. \[\lim_{x\to\infty}a^{x}= \begin{cases} 0 & , & \mathrm{if}-1<a<1 \\ 1 & , & \mathrm{if}a=1 \\ \infty & , & \mathrm{if}a>1 & \end{cases}\]

8. \[\lim_{x\to0}\frac{\log\left(1+\mathrm{k}x\right)}{x}=\mathrm{k},\mathrm{k}\in\mathrm{R}\]

1. \[\lim_{x\to0}\frac{\sin x}{x}=1=\lim_{x\to0}\frac{x}{\sin x}\]

2. $$\lim_{x\to0}\frac{\tan x}{x}=1=\lim_{x\to0}\frac{x}{\tan x}$$

3. \[\lim_{x\to0}\frac{\sin^{-1}x}{x}=1=\lim_{x\to0}\frac{x}{\sin^{-1}x}\]

4. \[\lim_{x\to0}\frac{\tan^{-1}x}{x}=1=\lim_{x\to0}\frac{x}{\tan^{-1}x}\]

5. \[\lim_{x\to0}\frac{\sin x^{\circ}}{x}=\frac{\pi}{180}\]

6. \[\lim_{x\to0}\cos x=1\]

7. \[\lim_{x\to0}\frac{\sin\mathrm{k}x}{x}=\lim_{x\to0}\frac{\tan\mathrm{k}x}{x}=\mathrm{k}\]

8. \[\lim_{x\to\infty}\frac{\sin x}{x}=\lim_{x\to\infty}\frac{\cos x}{x}=0\]

9. \[\lim_{x\to\infty}\frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x}}=1=\lim_{x\to\infty}\frac{\tan\left(\frac{1}{x}\right)}{\frac{1}{x}}\]

10. \[\lim_{x\to a}\frac{\sin\left(x-a\right)}{x-a}=1=\lim_{x\to a}\frac{\tan\left(x-a\right)}{x-a}\]

1. \[\lim_{x\to\mathrm{a}}x=\mathrm{a}\]

2. \[\lim_{x\to\mathrm{a}}x^\mathrm{n}=\mathrm{a}^\mathrm{n}\]

3. \[\lim_{x\to a}\mathrm{k}=\mathrm{k},\] (where k is a constatnt)

4. \[\lim_{x\to a}\sqrt[r]{x}=\sqrt[r]{a}\]

5. If P(x) is a polynomial, then \[\lim_{x\to a}\mathrm{P}(x)=\mathrm{P}(\mathrm{a})\]

6. \[\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}\]

7. \[\lim_{x\to\infty}\frac{1}{x^{k}}=0,\mathrm{where~k}>0\]

If f(x) ≤ g(x) ≤ h(x) and \[\lim_{x\to a}\mathrm{f}(x)=l=\lim_{x\to a}\mathrm{h}(x)\]

\[\therefore\lim_{x\to a}g(x)=l\]

If \[\lim_{x\to a}\frac{f(x)}{g(x)}\] is of the form \[\frac{0}{0}\mathrm{or}\frac{\infty}{\infty}\], then \[\lim_{x\to a}\frac{\mathrm{f}(x)}{\mathrm{g}(x)}=\lim_{x\to a}\frac{\mathrm{f}^{\prime}(x)}{\mathrm{g}^{\prime}(x)}\] Process is continued till the indeterminate form remains.