What is the limit as x approaches 0 of ((1+x)^8-1)/x ?

The limit as x approaches 0 of ((1+x)^8-1)/x is 8

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limit as x approaches 0 of ((1+x)^8-1)/x

x \to 0 lim (\frac{( 1 + x ) ^{8} - 1}{x})

8

Solution steps

x \to 0 lim (\frac{( 1 + x ) ^{8} - 1}{x})

Simplify

\frac{( 1 + x ) ^{8} - 1}{x} : (x^{2} + 2 x + 2 x + 2 + 2) (x^{2} + 2 x - 2 x + 2 - 2) (x^{2} + 2 x + 2) (x + 2)

= x \to 0 lim ((x^{2} + 2 x + 2 x + 2 + 2) (x^{2} + 2 x - 2 x + 2 - 2) (x^{2} + 2 x + 2) (x + 2))

Plug in the value

x = 0

= (0^{2} + 2 \cdot 0 + 2 \cdot 0 + 2 + 2) (0^{2} + 2 \cdot 0 - 2 \cdot 0 + 2 - 2) (0^{2} + 2 \cdot 0 + 2) (0 + 2)

Simplify

(0^{2} + 2 \cdot 0 + 2 \cdot 0 + 2 + 2) (0^{2} + 2 \cdot 0 - 2 \cdot 0 + 2 - 2) (0^{2} + 2 \cdot 0 + 2) (0 + 2) : 8

= 8

\int_{- 1}^{1} 3 (1 - x^{2}) d x

\int \frac{1}{h} d h

\int sec^{4} (x t a) n^{4} x d x

sin^{9} (x)

x \to \infty lim (\frac{3 ^{x + 1}}{3 ^{x}})

What is the limit as x approaches 0 of ((1+x)^8-1)/x ?
The limit as x approaches 0 of ((1+x)^8-1)/x is 8