Detailed Solution: Limits
Evaluate the following limit: $$ \lim_{x \to 0} \frac{5^x - 3^x}{4^x - 3^x} $$
Solution:
This takes the indeterminate form of $\frac{0}{0}$ when you plug in $x = 0$. We can solve it using the standard logarithmic limit formula: $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$.
Step 1: Divide the numerator and denominator by $x$.
To manipulate the expression into the standard formula, we divide both the top and the bottom by $x$:$$ \lim_{x \to 0} \frac{\frac{5^x - 3^x}{x}}{\frac{4^x - 3^x}{x}} $$
Step 2: Add and subtract $1$ in both the numerator and the denominator.
This allows us to group the terms properly:
- Numerator: $\frac{5^x - 1 - 3^x + 1}{x} = \frac{(5^x - 1) - (3^x - 1)}{x}$
- Denominator: $\frac{4^x - 1 - 3^x + 1}{x} = \frac{(4^x - 1) - (3^x - 1)}{x}$
So, our limit becomes:
$$ \lim_{x \to 0} \frac{\frac{5^x - 1}{x} - \frac{3^x - 1}{x}}{\frac{4^x - 1}{x} - \frac{3^x - 1}{x}} $$
Step 3: Apply the standard limit formula.
Now, we apply $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$ to each part of the fraction:
- Numerator limit: $\ln(5) - \ln(3)$
- Denominator limit: $\ln(4) - \ln(3)$
Step 4: Use logarithm properties to simplify.
Recall that $\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$.
- Numerator: $\ln\left(\frac{5}{3}\right)$
- Denominator: $\ln\left(\frac{4}{3}\right)$
Final Answer:
$$ \frac{\ln(5/3)}{\ln(4/3)} $$