Write the value of lim X - 0 f x - f x - x - c

Since we know that, d/x(f(x))=lim_h→ 0f(x+h) f(x)/h Let, h = x c and f(x) = c If h→ 0 then x→ c Therefore, d/dx(c)=lim_x→ ef(x+c x) f(c)/x c =lim_x→ ef(x) f(c ...

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Standard XII

Mathematics

Derivative from First Principle

Write the val...

Question

Write the value of $l i m_{x \to 0} \frac{f (x) - f (x)}{x - c}$

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Solution

Since we know that,

$\frac{d}{x} (f (x)) = l i m_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Let,

h = x - c and f(x) = c

If $h \to 0$ then $x \to c$

Therefore,

$\frac{d}{d x} (c) = l i m_{x \to e} \frac{f (x + c - x) - f (c)}{x - c}$

$= l i m_{x \to e} \frac{f (x) - f (c)}{x - c}$

$= f^{'} (c)$

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Write the value of limx→0f(x)−f(x)x−c

Since we know that, dx(f(x))=limh→0f(x+h)−f(x)h Let, h = x - c and f(x) = c If h→0 then x→c Therefore, ddx(c)=limx→ef(x+c−x)−f(c)x−c =limx→ef(x)−f(c)x−c =f′(c)

Write the value of $l i m_{x \to 0} \frac{f (x) - f (x)}{x - c}$

Since we know that,

$\frac{d}{x} (f (x)) = l i m_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Let,

h = x - c and f(x) = c

If $h \to 0$ then $x \to c$

Therefore,

$\frac{d}{d x} (c) = l i m_{x \to e} \frac{f (x + c - x) - f (c)}{x - c}$

$= l i m_{x \to e} \frac{f (x) - f (c)}{x - c}$

$= f^{'} (c)$