Solved example of limits by rationalizing
Simplifying
Plug in the value $0$ into the limit
Add the values $5$ and $0$
Calculate the power $\sqrt{5}$
Subtract the values $2.2361$ and $-2.2361$
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 derivative of the constant function ($-\sqrt{5}$) is equal to zero
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
The derivative of the constant function ($5$) is equal to zero
The derivative of the linear function is equal to $1$
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
The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$
Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number
Evaluate the limit $\lim_{x\to0}\left(\frac{1}{\sqrt{5+x}}\right)$ by replacing all occurrences of $x$ by $0$
Add the values $5$ and $0$
Calculate the power $\sqrt{5}$
Divide $1$ by $\sqrt{5}$
Multiply $\frac{1}{2}$ times $\frac{\sqrt{5}}{5}$
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