Exponentials Vs. Polynomials In A Limit

Calculus Level 1

$\large \lim_{x\to0} \dfrac{3^x-2^x} x$

If the value of the limit above can be expressed as $\ln \left( \dfrac ab\right)$ , where $a$ and $b$ are coprime positive integers, find $a+b$ .

The answer is 5.

2 solutions

Rishabh Jain
Mar 1, 2016

Apply (1). L'hopital Rule or $\color{#D61F06}{\text{(2).Use }\lim_{x\to 0}\dfrac{a^x-1}{x}=\ln a}$ Given limit can be written as: $\large\mathfrak{L}= \lim_{x\to0} \dfrac{3^x-1} x- \lim_{x\to0} \dfrac{2^x-1} x$ $\large =\ln 3- \ln 2=\ln(\dfrac 32)$ $\huge \therefore ~ 2+3=\boxed{\color{#007fff}{5}}$

Sandeep Bhardwaj
Mar 1, 2016

HINT: Use L'hopital's Rule to solve the given limits .

You are absolutely right

Adarsh Mahor - 5 years, 3 months ago

Exponentials Vs. Polynomials In A Limit

The answer is 5.

2 solutions

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