Question

$\lim_{x \rightarrow 1} \frac{ab^x - a^x b}{x^2 - 1} = $

• A. $-\frac{ab}{2} \log\left(\frac{b}{a}\right)$
• B. $\frac{ab}{2} \log\left(\frac{b}{a}\right)$
• C. $ab \log\left(\frac{b}{a}\right)$
• D. $-ab \log\left(\frac{b}{a}\right)$

Hint:

Whenever a calculus limit produces a $\frac{0}{0}$ or $\frac{\infty}{\infty}$ format, apply L'Hospital's rule immediately. Differentiating variables independently bypasses tedious algebraic grouping and leads directly to the answer in a single step!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are asked to evaluate the limit of the rational exponential function $\frac{ab^x - a^x b}{x^2 - 1}$ as $x$ approaches $1$.

Step 2: Key Formula or Approach:
Let's first check the form of the limit by substituting $x = 1$: $$\text{Numerator} = ab^1 - a^1b = ab - ab = 0$$ $$\text{Denominator} = 1^2 - 1 = 0$$ Since this yields an indeterminate $\frac{0}{0}$ form, we can apply L'Hospital's Rule, which states that if a limit has an indeterminate form $\frac{0}{0}$, it can be evaluated by taking the derivative of the numerator and denominator separately: $$\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}$$ Recall the standard derivative formula for exponential terms: $\frac{d}{dx}(k^x) = k^x \log k$.

Step 3: Detailed Explanation:
Apply differentiation to the numerator and denominator with respect to $x$: $$\text{Derivative of Numerator} = \frac{d}{dx}(ab^x - ba^x) = a(b^x \log b) - b(a^x \log a)$$ $$\text{Derivative of Denominator} = \frac{d}{dx}(x^2 - 1) = 2x$$ Now, rewrite our limit expression using these derived components: $$L = \lim_{x \rightarrow 1} \frac{ab^x \log b - ba^x \log a}{2x}$$ Since this new form is determinate, substitute $x = 1$ directly into the expression: $$L = \frac{ab^1 \log b - ba^1 \log a}{2(1)}$$ $$L = \frac{ab \log b - ab \log a}{2}$$ Factor out the common term $ab$ from the numerator: $$L = \frac{ab}{2} (\log b - \log a)$$ Using the logarithmic subtraction property $\log b - \log a = \log\left(\frac{b}{a}\right)$, we can simplify the expression to its final form: $$L = \frac{ab}{2} \log\left(\frac{b}{a}\right)$$

Step 4: Final Answer:
The limit values resolve cleanly to $\frac{ab}{2} \log\left(\frac{b}{a}\right)$, which matches option (B).