What is the limit as x approaches 0 of (sqrt(x^2+a^2)+a)/(sqrt(x^2+b^2)+b) ?

The limit as x approaches 0 of (sqrt(x^2+a^2)+a)/(sqrt(x^2+b^2)+b) is 0

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limit as x approaches 0 of (sqrt(x^2+a^2)+a)/(sqrt(x^2+b^2)+b)

x \to 0 lim (\frac{x ^{2} + a ^{2} + a}{x ^{2} + b ^{2} + b})

0

Solution steps

x \to 0 lim (\frac{x ^{2} + a ^{2} + a}{x ^{2} + b ^{2} + b})

Multiply by the conjugate of

x^{2} + a^{2} + a : \frac{x ^{2}}{x ^{2} + a ^{2} - a}

= x \to 0 lim \frac{\frac{x ^{2}}{x ^{2} + a ^{2} - a}}{x ^{2} + b ^{2} + b}

Plug in the value

x = 0

= \frac{\frac{0 ^{2}}{0 ^{2} + a ^{2} - a}}{0 ^{2} + b ^{2} + b}

Simplify

\frac{\frac{0 ^{2}}{0 ^{2} + a ^{2} - a}}{0 ^{2} + b ^{2} + b} : 0

= 0

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

x \to 5 lim (\frac{x - 5}{x + 1})

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

x \to 5 lim (x^{2} - 6)

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

What is the limit as x approaches 0 of (sqrt(x^2+a^2)+a)/(sqrt(x^2+b^2)+b) ?
The limit as x approaches 0 of (sqrt(x^2+a^2)+a)/(sqrt(x^2+b^2)+b) is 0