Get the solution to "How can we find limy→{b}y−bln⁡{y}−ln⁡{b}"

Caerswso1pc

Answered question

2022-03-25

How can we find

lim_{y \to {b}} \frac{y - b}{\ln {y} - \ln {b}}

Jesse Gates

Beginner2022-03-26Added 19 answers

Let us look at the limit in the title and the definition of

\log x

(please excuse me for using

\log

notation instead of

\ln

) as in your comment,

\log x = \int_{1}^{x} \frac{1}{t} d t

By the fundamental theorem of calculus we have

\frac{d}{d x} \log x = \frac{1}{x}

on the other hand

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = lim_{y \to x} \frac{f (y) - f (x)}{y - x}

hence

lim_{y \to x} \frac{y - x}{\log y - \log x} = lim_{y \to x} \frac{1}{\frac{\log y - \log x}{y - x}} = \frac{1}{\frac{1}{x}} = x

where in the last step we used the quotient rule

lim_{y \to a}, g (y) = A, and, lim_{y \to a}, h (y) = B \neq 0 implies lim_{y \to a}, \frac{g (y)}{h (y)} = \frac{A}{B}

with

g (y) = 1

Jeffrey Jordon

Expert2022-07-05Added 2605 answers

Answer is given below (on video)

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