Here, we show you a step-by-step solved example of limits by rationalizing. This solution was automatically generated by our smart calculator:
Plug in the value $0$ into the limit
Add the values $5$ and $0$
Cancel like terms $\sqrt{5}$ and $-\sqrt{5}$
If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{\sqrt{5+x}-\sqrt{5}}{x}\right)$ as $x$ tends to $0$, we can see that it gives us an indeterminate form
We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately
Find the derivative of the numerator
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
Find the derivative of the denominator
The derivative of the linear function is equal to $1$
Any expression divided by one ($1$) is equal to that same expression
After deriving both the numerator and denominator, the limit results in
Multiplying the fraction by $\left(5+x\right)^{-\frac{1}{2}}$
Any expression multiplied by $1$ is equal to itself
Multiplying the fraction by $\left(5+x\right)^{-\frac{1}{2}}$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Evaluate the limit $\lim_{x\to0}\left(\frac{1}{2\sqrt{5+x}}\right)$ by replacing all occurrences of $x$ by $0$
Add the values $5$ and $0$
Evaluate the limit $\lim_{x\to0}\left(\frac{1}{2\sqrt{5+x}}\right)$ by replacing all occurrences of $x$ by $0$
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