Solved example of base change formula of logarithms
Plug in the value $0$ into the limit
Calculate the power $5^0$
Subtract the values $1$ and $-1$
Add the values $1$ and $0$
Calculating the natural logarithm of $1$
If we directly evaluate the limit $\lim_{x\to 0}\left(\frac{5^x-1}{\ln\left(1+x\right)}\right)$ 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 ($-1$) is equal to zero
Applying the derivative of the exponential function
The derivative of the linear function is equal to $1$
Calculating the natural logarithm of $5$
Find the derivative of the denominator
The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$
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 ($1$) is equal to zero
The derivative of the linear function is equal to $1$
Divide fractions $\frac{\ln\left(5\right)5^x}{\frac{1}{1+x}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
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: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$
Evaluate the limit $\lim_{x\to0}\left(5^x\left(1+x\right)\right)$ by replacing all occurrences of $x$ by $0$
Add the values $1$ and $0$
Calculate the power $5^0$
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