limit as h(x) approaches 0 of (f(x)(x+2h(x))-f(x))/(h(x))

解答

h (x) \to 0 lim (\frac{f ( x ) ( x + 2 h ( x ) ) - f ( x )}{h ( x )})

未定义

求解步骤

h (x) \to 0 lim (\frac{f ( x ) ( x + 2 h ( x ) ) - f ( x )}{h ( x )})

f (x) (x + 2 h (x)) - f (x) = f (x) (x + 2 h (x)) (1 - \frac{f ( x )}{f ( x ) ( x + 2 h ( x ) )})

= h (x) \to 0 lim \frac{f ( x ) ( x + 2 h ( x ) ) ( 1 - \frac{f ( x )}{f ( x ) ( x + 2 h ( x ) )} )}{h ( x )}

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= f (x) \cdot h (x) \to 0 lim \frac{( x + 2 h ( x ) ) ( 1 - \frac{f ( x )}{f ( x ) ( x + 2 h ( x ) )} )}{h ( x )}

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

不定型除外

= f (x) \frac{lim _{h (x) \to 0} ( ( x + 2 h ( x ) ) ( 1 - \frac{f ( x )}{f ( x ) ( x + 2 h ( x ) )} ) )}{lim _{h (x) \to 0} ( h ( x ) )}

h (x) \to 0 lim ((x + 2 h (x)) (1 - \frac{f ( x )}{f ( x ) ( x + 2 h ( x ) )})) = x - 1

h (x) \to 0 lim (h (x)) = 0

= f (x) \frac{x - 1}{0}

\frac{x - 1}{0}

不在定义域

= 未定义

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