Evaluate the following:

1. $\displaylines{lim \\ x \to 0} \dfrac{sinax}{x}$

Solution:
Given function takes indeterminate (0/0) form at x = 0. So,

$\displaylines{lim \\  x \to 0} \dfrac{sinax}{x} * \dfrac{a}{a}$

$= a * \displaylines{lim \\ x \to 0} \dfrac{sinax}{ax}$

[$\because \displaylines{lim \\x \to 0} \dfrac{sinx}{x} = 1$]

$= a * 1$

$= a$


2. $\displaylines{lim \\ x \to 0} \dfrac{tan bx}{x}$

Solution:
Given function takes indeterminate (0/0) form at x = 0. So,

$\displaylines{lim \\ x \to 0} \dfrac{sinbx}{cosbx} * \dfrac{1}{x}$

$= \displaylines{lim \\x \to 0} \dfrac{sinbx}{cosbx} * \dfrac{b}{b} * \dfrac{1}{x}$

$= b * \displaylines{lim \\x \to 0} \dfrac{sinbx}{bx} * \dfrac{1}{cosbx}$

[$\because \displaylines{lim \\x \to 0} \dfrac{sinx}{x} = 1$]

$ = b * 1 * \dfrac{1}{cos0}$

$= b * 1 * \dfrac{1}{1}$

$= b$


3. $\displaylines{lim \\ x \to 0} \dfrac{\sin mx}{\sin nx}$

Solution:
Given function takes indeterminate (0/0) form at x = 0. So,

$\displaylines{lim \\ x \to 0} \dfrac{\sin mx}{\sin nx} * \dfrac{mx}{mx} * \dfrac{nx}{nx}$

$= \displaylines{lim \\ x \to 0} \dfrac{\sin mx}{mx} * \dfrac{m}{n} * \dfrac{1}{\frac{\sin nx}{nx} }$

$= \dfrac{m}{n} * \displaylines{lim \\ x \to 0} \dfrac{\sin mx}{mx} * \dfrac{1}{\frac{\sin nx}{nx}}$

[$\because \displaylines{lim \\x \to 0} \dfrac{sinx}{x} = 1$]

$= \dfrac{m}{n} * 1 * \dfrac{1}{1}$

$ = \dfrac{m}{n}$


4. $\displaylines{lim \\ x \to 0} \dfrac{tan ax}{tan bx}$

Solution:
Given function takes indeterminate (0/0) form at x = 0. So,

$\displaylines{lim \\ x \to 0} \dfrac{sinax}{cosax} * \dfrac{cos bx}{sin bx}$

$= \displaylines{lim \\x \to 0}\dfrac{sinax}{sinbx} * \dfrac{cosbx}{cosax}$

$= \displaylines{lim \\ x \to 0}\dfrac{sinax}{sinbx} * \displaylines{lim \\x \to 0}\dfrac{cosbx}{cosax}$

$= \displaylines{lim \\x \to 0} \dfrac{sinax}{ax} * \dfrac{1}{\frac{sin bx}{bx}} * \dfrac{ax}{bx} * \dfrac{cos0}{cos0}$

$= \displaylines{lim \\ x \to 0} \dfrac{sinax}{ax} * \dfrac{1}{\frac{sin bx}{bx}} * \dfrac{a}{b} * \dfrac{1}{1}$

[$\because \displaylines{lim \\x \to 0} \dfrac{sinx}{x} = 1$]

$= 1 * \dfrac{1}{1} * \dfrac{a}{b} * 1$

$= \dfrac{a}{b}$


5. $\displaylines{lim \\ x \to 0} \dfrac{\sin px}{\tan qx}$

Solution:

Here,
The function takes indeterminate (0/0) form at x = 0.

So,
$\displaylines{lim \\ x \to 0} (\sin px ) * \dfrac{ \cos qx}{\sin qx}$

$= \displaylines{lim \\x \to 0}\dfrac{\sin px{{px} * (\cos qx) * \dfrac{1}{\dfrac{\sin qx}{qx}} * \dfrac{px}{qx}$

$= \displaylines{lim \\x \to 0} \dfrac{\sin px}{px} * (\cos qx) * \dfrac{1}{\dfrac{\sin qx}{qx}} * \dfrac{p}{q}$

[$\because \displaylines{lim \\x \to 0} \dfrac{sinx}{x} = 1$]

$= 1 * cos 0 * \dfrac{1}{1} * \dfrac{p}{q}$

$= \dfrac{p}{q}$


6. $\displaylines{lim \\ x \to a} \dfrac{sin ( x-a)}{x^2 - a^2}$

Solution:

Given,

$\displaylines{lim \\ x \to a} \dfrac{sin ( x-a)}{x^2 - a^2}$

$= \displaylines{lim \\ x - a \to a - a} \dfrac{sin(x-a}{(x-a)} × \displaylines{lim \\ x \to a} \dfrac{1}{(x+a)}$

$= \displaylines{lim \\ x- a \to 0} \dfrac{sin(x-a)}{(x-a)} × \dfrac{1}{(a+a)}$

$= 1 × \dfrac{1}{2a}$

$= \dfrac{1}{2a}$



7. $\displaylines{lim \\x \to p} \dfrac{x^2 - p^2}{tan (x - p)}$

Solution:

$\displaylines{lim \\x \to p} \dfrac{x^2 - p^2}{tan (x - p)}$

$= \displaylines{lim \\ x \to p} \dfrac{(x+p)(x-p)}{tan(x-p)}$

$= \displaylines{lim \\ x - p \to p -p} \dfrac{(x-p)}{tan (x -p)} × \displaylines{lim \\ x \to p} (x +p)$

$= \displaylines{lim \\ x - p \to 0} \dfrac{1}{ \frac{tan(x-p)}{(x-p)}} × (p+p)$

$= \dfrac{1}{1} × (2p)$

$= 2p$


8. $\displaylines{lim \\x \to 0} \dfrac{\sin ax * \cos bx}{\sin cx}$

Solution:

$\displaylines{lim \\ x \to 0} \dfrac{\sin ax * \cos bx}{\sin cx}$

$= \displaylines{lim \\ x \to 0} \dfrac{\sin \frac{ax}{ax} × ax × \cos bx}{\sin \frac{cx}{cx} × cx}$

$= 1 × ax × cos(b×0) × \dfrac{1}{1 × cx}$

$= ax × 1 × \dfrac{1}{cx}$

$= \dfrac{a}{c}$


9. $\displaylines{lim \\x \to 0}\dfrac{1 - cosx}{x^2}$

Solution:

Given,

$\displaylines{lim \\x \to 0}\dfrac{1 - cosx}{x^2}$

$= \displaylines{lim \\ x \to 0} \dfrac{2sin^2 \frac{x}{2}}{x^2}$

$= 2× \displaylines{lim \\ x \to 0} \dfrac{sin^2 \frac{x}{2}}{ \left ( \frac{x}{2} \right)^2} × \dfrac{1}{2^2}$

$= 2× 1 × \dfrac{1}{2^2}$

$= \dfrac{1}{2}$


10. $\displaylines{lim \\x \to 0}\dfrac{1- \cos 6x}{x^2}$

Solution:

Given,

$\displaylines{lim \\x \to 0}\dfrac{1- \cos 6x}{x^2}$

$= \displaylines{ lim \\x \to 0} \dfrac{2 sin^2 3x}{x^2}$

$= 2× \displaylines{lim \\x \to 0} \dfrac{sin^2 3x}{(3x)^2} × 3^2$

$= 2 × 1 × 9$

$= 18$


29. Find the limits of:

a) $\displaylines{lim \\ x \to 0} \dfrac{e^{6x} - 1}{x}$

Solution:

Given function takes (0/0) indeterminate form at x = 0 so,

$\displaylines{lim \\ x \to 0} \dfrac{e^{6x} - 1}{x}$

$= \displaylines{lim \\ x \to 0} \dfrac{e^{6x} - 1}{6x} * 6$

$= 6 * \displaylines{lim \\ x \to 0} \dfrac{e^{6x} - 1}{6x}$

$= 6 * 1$

[$ \because \displaylines{lim \\ x \to 0} \dfrac{e^x -1}{x} = 1$]

$= 6$


b) $\displaylines{lim \\ x \to 0} \dfrac{e^{2x} - 1}{x . 2^{x+1}}$

Solution:

Given function takes (0/0) indeterminate form at x = 0

$\displaylines{lim \\ x \to 0} \dfrac{e^{2x} - 1}{x . 2^{x+1}}$

$= \displaylines{ lim \\ x \to 0} \dfrac{e^{2x} -1 }{x * 2^x * 2^1}$

$= \displaylines{lim \\ x \to 0} \ left [ \dfrac{e^{2x} - 1}{2x } * \dfrac{1}{2^x} \right ]$

[$ \because \displaylines{lim \\ x \to 0} \dfrac{e^x -1}{x} = 1$]

$= 1 * \dfrac{1}{2^0}$

$= 1 * 1$

$= 1$


c) $\displaylines{ lim \\ x \to 0} \dfrac{e^{ax} - e^{bx}}{x}$

Solution:

Given function takes (0/0) indeterminate form at x = 0

$\displaylines{ lim \\ x \to 0} \dfrac{e^{ax} - e^b{x} + 1 - 1}{x}$

$= \displaylines{ lim \\ x \to 0} \dfrac{(e^{ax} - 1) - (e^{bx} -1)}{x}$

$= \displaylines{ lim \\ x \to 0} \dfrac{e^{ax}-1}{ax}*a - \displaylines{ lim \\ x \to 0} \dfrac{e^{bx} -1}{bx} * b$ 

$= (1 * a) - (1 * b)$

$= a - b$