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

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

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limit as h approaches 0 of (sqrt(x^2+2xh+h^2+1))/h-(sqrt(x^2+1))/h

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

\frac{x}{x ^{2} + 1}

Solution steps

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

Multiply by the conjugate of

\frac{x ^{2} + 2 x h + h ^{2} + 1}{h} - \frac{x ^{2} + 1}{h} : \frac{\frac{2 x}{h} + 1}{\frac{x ^{2} + 2 x h + h ^{2} + 1}{h} + \frac{x ^{2} + 1}{h}}

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

Simplify

\frac{\frac{2 x}{h} + 1}{\frac{x ^{2} + 2 x h + h ^{2} + 1}{h} + \frac{x ^{2} + 1}{h}} : \frac{2 x + h}{x ^{2} + 2 x h + h ^{2} + 1 + x ^{2} + 1}

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

Plug in the value

h = 0

= \frac{2 x + 0}{x ^{2} + 2 x \cdot 0 + 0 ^{2} + 1 + x ^{2} + 1}

Simplify

\frac{2 x + 0}{x ^{2} + 2 x \cdot 0 + 0 ^{2} + 1 + x ^{2} + 1} : \frac{x}{x ^{2} + 1}

= \frac{x}{x ^{2} + 1}

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