What is the limit as k approaches 0 of (x^2k+3xk^2+k^2)/(2xk+5k^2) ?

The limit as k approaches 0 of (x^2k+3xk^2+k^2)/(2xk+5k^2) is x/2

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limit as k approaches 0 of (x^2k+3xk^2+k^2)/(2xk+5k^2)

k \to 0 lim (\frac{x ^{2} k + 3 x k ^{2} + k ^{2}}{2 x k + 5 k ^{2}})

\frac{x}{2}

Solution steps

k \to 0 lim (\frac{x ^{2} k + 3 x k ^{2} + k ^{2}}{2 x k + 5 k ^{2}})

Simplify

\frac{x ^{2} k + 3 x k ^{2} + k ^{2}}{2 x k + 5 k ^{2}} : \frac{x ^{2} + 3 k x + k}{2 x + 5 k}

= k \to 0 lim (\frac{x ^{2} + 3 k x + k}{2 x + 5 k})

Plug in the value

k = 0

= \frac{x ^{2} + 3 \cdot 0 \cdot x + 0}{2 x + 5 \cdot 0}

Simplify

\frac{x ^{2} + 3 \cdot 0 \cdot x + 0}{2 x + 5 \cdot 0} : \frac{x}{2}

= \frac{x}{2}

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What is the limit as k approaches 0 of (x^2k+3xk^2+k^2)/(2xk+5k^2) ?
The limit as k approaches 0 of (x^2k+3xk^2+k^2)/(2xk+5k^2) is x/2