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Chapter 8: Problem 10

Evaluate \(L^{\prime}(1)\) by using the sequence \((1+1 / n)\) and the fact that \(e=\lim \left((1+1 / n)^{n}\right)\).

Short Answer

Expert verified
The value of \(L^{\prime}(1)\) is 1.

Step by step solution

01

Define the function and the derivative

Let's define a function \(L(x) = \ln(x)\), where \(\ln(x)\) is the natural logarithm function. The derivative \(L^{\prime}(x)\) of this function is \(1/x\). Therefore, \(L^{\prime}(1) = 1/1 = 1\).
02

Relate the sequence to the limit definition of the derivative

The sequence \((1+1/n)^n\) approaches \(e\) as \(n\) tends towards infinity. To find the derivative at \(x=1\), we need to calculate the limit as \(n\) tends towards infinity: \(\lim_{n \to \infty} \dfrac{L(1+1/n) - L(1)}{1/n}\).
03

Evaluate the limit

By plugging in the definition of \(L(x)\) from Step 1 into our formula and simplifying, we find that this limit is equal to \(\lim_{n \to \infty} \dfrac{\ln{(1 + 1/n)} - \ln{1}}{1/n} = \lim_{n \to \infty} n \cdot \ln{(1 + 1/n)} = \lim_{n \to \infty} \ln{((1 + 1/n)^n)}\). Fuurthermore, because \(\lim_{n \to \infty} (1+1/n)^n\) equals to \(e\), we then have that \(\lim_{n \to \infty} \ln{((1 + 1/n)^n)} = \ln{e} = 1\)

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Most popular questions from this chapter

Let \(f_{n}(x):=1\) for \(x \in(0,1 / n)\) and \(f_{n}(x):=0\) elsewhere in \([0,1]\). Show that \(\left(f_{n}\right)\) is a decreasing sequence of discontinuous functions that converges to a continuous limit function. but the convergence is not uniform on \([0,1]\).

Let \(\left(f_{n}\right)\) be a sequence of functions that converges unjformly to \(f\) on \(A\) and that satisfies \(\left|f_{n}(x)\right| \leq M\) for all \(n \in \mathbb{N}\) and all \(x \in A\). If \(g\) is continuous on the interval \([-M, M]\), show that the sequence \(\left(g \circ f_{n}\right)\) converges uniformly to \(g \circ f\) on \(A\).

Show that if \(\left(f_{n}\right),\left(g_{n}\right)\) converge uniformly on the set \(A\) to \(f, g\), respectively, then \(\left(f_{n}+g_{n}\right)\) converges uniformly on \(A\) to \(f+g\).

Let \(f_{n}(x):=x / n\) for \(x \in[0, \infty), n \in \mathbb{N}\). Show that \(\left(f_{n}\right)\) is a decreasing sequence of continuous functions that converges to a continuous limit function, but the convergence is not uniform on \([0 . \infty)\)

Let \(g_{n}(x):=e^{-n x} / n\) for \(x \geq 0, n \in \mathbb{N}\). Examine the relation between \(\lim \left(g_{n}\right)\) and \(\lim \left(g_{n}^{\prime}\right)\).
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