What is the limit as h approaches 0 of ((8(1/2+h)^8-8(1/2)^8))/h ?

The limit as h approaches 0 of ((8(1/2+h)^8-8(1/2)^8))/h is 1/2

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limit as h approaches 0 of ((8(1/2+h)^8-8(1/2)^8))/h

h \to 0 lim \frac{( 8 ( \frac{1}{2} + h ) ^{8} - 8 ( \frac{1}{2} ) ^{8} )}{h}

\frac{1}{2}

Decimal

0.5

Solution steps

h \to 0 lim (\frac{8 ( \frac{1}{2} + h ) ^{8} - 8 ( \frac{1}{2} ) ^{8}}{h})

Simplify

\frac{8 ( \frac{1}{2} + h ) ^{8} - 8 ( \frac{1}{2} ) ^{8}}{h} : \frac{256 ( \frac{1}{2} + h ) ^{8} - 1}{32 h}

= h \to 0 lim (\frac{256 ( \frac{1}{2} + h ) ^{8} - 1}{32 h})

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

= \frac{1}{32} \cdot h \to 0 lim (\frac{256 ( \frac{1}{2} + h ) ^{8} - 1}{h})

Factor out

256 (\frac{1}{2} + h)^{8}

from

256 (\frac{1}{2} + h)^{8} - 1 : 256 (\frac{1}{2} + h)^{8} (1 - \frac{1}{256 ( \frac{1}{2} + h ) ^{8}})

= \frac{1}{32} \cdot h \to 0 lim \frac{256 ( \frac{1}{2} + h ) ^{8} ( 1 - \frac{1}{256 ( \frac{1}{2} + h ) ^{8}} )}{h}

Simplify

\frac{256 ( \frac{1}{2} + h ) ^{8} ( 1 - \frac{1}{256 ( \frac{1}{2} + h ) ^{8}} )}{h} : 2 (4 h^{2} + 4 h + 22 h + 2 + 2) (4 h^{2} + 4 h - 22 h + 2 - 2) (4 h^{2} + 4 h + 2) (2 h + 2)

= \frac{1}{32} \cdot h \to 0 lim (2 (4 h^{2} + 4 h + 22 h + 2 + 2) (4 h^{2} + 4 h - 22 h + 2 - 2) (4 h^{2} + 4 h + 2) (2 h + 2))

Plug in the value

h = 0

= \frac{1}{32} \cdot 2 (4 \cdot 0^{2} + 4 \cdot 0 + 22 \cdot 0 + 2 + 2) (4 \cdot 0^{2} + 4 \cdot 0 - 22 \cdot 0 + 2 - 2) (4 \cdot 0^{2} + 4 \cdot 0 + 2) (2 \cdot 0 + 2)

Simplify

\frac{1}{32} \cdot 2 (4 \cdot 0^{2} + 4 \cdot 0 + 22 \cdot 0 + 2 + 2) (4 \cdot 0^{2} + 4 \cdot 0 - 22 \cdot 0 + 2 - 2) (4 \cdot 0^{2} + 4 \cdot 0 + 2) (2 \cdot 0 + 2) : \frac{1}{2}

= \frac{1}{2}

a \to - 2 lim (\frac{a ^{3} + 8}{a + 2})

x \to 0 + lim (3 x + 2)

x \to 3 lim ((x - 3) csc (π x))

x \to - \frac{1}{2} lim (x)

n \to \infty lim ((\frac{n + 1}{n + 2})^{n + 2})

What is the limit as h approaches 0 of ((8(1/2+h)^8-8(1/2)^8))/h ?
The limit as h approaches 0 of ((8(1/2+h)^8-8(1/2)^8))/h is 1/2