What is the limit as x approaches 0 of ((1+x)^6-1)/x ?

The limit as x approaches 0 of ((1+x)^6-1)/x is 6

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limit as x approaches 0 of ((1+x)^6-1)/x

x \to 0 lim (\frac{( 1 + x ) ^{6} - 1}{x})

6

Solution steps

x \to 0 lim (\frac{( 1 + x ) ^{6} - 1}{x})

Simplify

\frac{( 1 + x ) ^{6} - 1}{x} : (x + 2) ((x + 1)^{4} + (x + 1)^{2} + 1)

= x \to 0 lim ((x + 2) ((x + 1)^{4} + (x + 1)^{2} + 1))

Plug in the value

x = 0

= (0 + 2) ((0 + 1)^{4} + (0 + 1)^{2} + 1)

Simplify

(0 + 2) ((0 + 1)^{4} + (0 + 1)^{2} + 1) : 6

= 6

What is the limit as x approaches 0 of ((1+x)^6-1)/x ?
The limit as x approaches 0 of ((1+x)^6-1)/x is 6