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Question

$\lim _{x \rightarrow 0} \frac{x \cdot 2^{x}}{2^{x}-1}=$ (A) $\ln 2$ (B) 1 (C) 2 (D) $\frac{1}{\ln 2}$

   $\lim _{x \rightarrow 0} \frac{x \cdot 2^{x}}{2^{x}-1}=$
(A) $\ln 2$
(B) 1
(C) 2
(D) $\frac{1}{\ln 2}$

Instant Answer

Solved by Expert Amrita Bhasin

08/28/2019

Step 1

If we plug in $x = 0$, we get an indeterminate form of $0/0$. This is a signal that we can apply L'Hopital's rule. Show more…

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$\lim _{x \rightarrow 0} \frac{x \cdot 2^{x}}{2^{x}-1}=$ (A) $\ln 2$ (B) 1 (C) 2 (D) $\frac{1}{\ln 2}$

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Key Concepts

Limits

Limits are fundamental in calculus and are used to study the behavior of functions as the input approaches a particular value. They help in understanding the continuity, differentiability, and the long-term behavior of functions, and are essential for defining derivatives and integrals.

Indeterminate Forms

Indeterminate forms, such as 0/0, occur when evaluating a limit leads to an unclear or undefined expression. Recognizing these forms is crucial because they indicate that further analysis, often through algebraic manipulation or special techniques, is required to determine the actual limit.

Exponential Functions

Exponential functions are functions of the form f(x) = a^x, where the base a is a positive constant. They exhibit distinctive growth or decay properties and play a significant role in various areas of mathematics, including calculus, complex analysis, and differential equations.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms like 0/0 or ?/?. By taking the derivatives of the numerator and the denominator separately, the rule simplifies the process of finding the limit, provided certain conditions are met.

Taylor Series Expansion

Taylor series expansion is a technique used to approximate functions near a specific point by expressing them as an infinite sum of terms calculated from the function's derivatives at that point. This method is particularly useful when dealing with limits or when simplifying complex functions into more manageable polynomial forms.

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