What is the limit as x approaches 0 of ((1+x)^7-1)/x ?

The limit as x approaches 0 of ((1+x)^7-1)/x is 7

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limit as x approaches 0 of ((1+x)^7-1)/x

x \to 0 lim (\frac{( 1 + x ) ^{7} - 1}{x})

7

Solution steps

x \to 0 lim (\frac{( 1 + x ) ^{7} - 1}{x})

Simplify

\frac{( 1 + x ) ^{7} - 1}{x} : x + (x + 1)^{6} + (x + 1)^{5} + (x + 1)^{4} + (x + 1)^{3} + (x + 1)^{2} + 2

= x \to 0 lim (x + (x + 1)^{6} + (x + 1)^{5} + (x + 1)^{4} + (x + 1)^{3} + (x + 1)^{2} + 2)

Plug in the value

x = 0

= 0 + (0 + 1)^{6} + (0 + 1)^{5} + (0 + 1)^{4} + (0 + 1)^{3} + (0 + 1)^{2} + 2

Simplify

0 + (0 + 1)^{6} + (0 + 1)^{5} + (0 + 1)^{4} + (0 + 1)^{3} + (0 + 1)^{2} + 2 : 7

= 7

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What is the limit as x approaches 0 of ((1+x)^7-1)/x ?
The limit as x approaches 0 of ((1+x)^7-1)/x is 7