Limit of (a^x-1)/x
In this tutorial we shall discuss another very important formula of limits, \[\mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \ln a\]
Let us consider the relation \[\mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x}\]
Let $$y = {a^x} – 1$$, then $$1 + y = {a^x}$$, we have
Consider the relation \[1 + y = {a^x}\]
Using the logarithm on both sides, we have \[\begin{gathered} \ln \left( {1 + y} \right) = \ln {a^x} \\ \Rightarrow \ln \left( {1 + y} \right) = x\ln a \\ \Rightarrow x = \frac{{\ln \left( {1 + y} \right)}}{{\ln a}} \\ \end{gathered} \]
Also $$\mathop {\lim }\limits_{x \to 0} y = \mathop {\lim }\limits_{x \to 0} \left( {{a^x} – 1} \right) = {a^0} – 1 = 1 – 1 = 0$$.
This shows that $$y \to 0$$ as $$x \to 0$$. Therefore, the given limit can be written as \[\begin{gathered} \mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \mathop {\lim }\limits_{y \to 0} \frac{y}{{\frac{{\ln \left( {1 + y} \right)}}{{\ln a}}}} \\ \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \mathop {\lim }\limits_{y \to 0} \frac{{\ln a}}{{\frac{1}{y}\ln \left( {1 + y} \right)}} \\ \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \mathop {\lim }\limits_{y \to 0} \frac{{\ln a}}{{\ln {{\left( {1 + y} \right)}^{\frac{1}{y}}}}} \\ \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \frac{{\ln a}}{{\mathop {\lim }\limits_{y \to 0} \ln {{\left( {1 + y} \right)}^{\frac{1}{y}}}}} \\ \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \frac{{\ln a}}{{\ln \left[ {\mathop {\lim }\limits_{y \to 0} {{\left( {1 + y} \right)}^{\frac{1}{y}}}} \right]}} \\ \end{gathered} \]
Using the relation $$\mathop {\lim }\limits_{x \to 0} {\left( {1 + x} \right)^{\frac{1}{x}}} = e$$, we have \[ \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \frac{{\ln a}}{{\ln e}}\]
But $$\ln e = 1$$, we have \[ \Rightarrow \mathop {\lim }\limits_{x \to 0} \frac{{{a^x} – 1}}{x} = \frac{{\ln a}}{1} = \ln a\]
Gallery
Strategy In Finding Limits Article Khan Academy
Solved Use Lim X Rightarrow 0 Sinx X 1 To Evaluate Th
Limit At Infinity Problems And Solutions Matheno Com
How Can E X Seemingly Approach 0 As X Approaches Negative
Math 130trig Limits Day 10
2 2 The Limit Of A Function Calculus Volume 1
View Question What Does Lim X Goes To 0 Mean
What Is The Solution Of Lim X 0 Tanx 2 X Quora
Limit Calculator Wolfram Alpha
Lim X 0 E X E Sinx X Sinx
Limit At Infinity Problems And Solutions Matheno Com
Practice Limits Of Functions Brilliant
Lim X 0 3sin2x 2x 3x 2 Tan3x Math Limits And
How Do You Find The Limit Of Sqrt 1 H 1 H As H Approaches
Pdf In Praise Of Y X Alpha Sin 1 X
L Hopital S Rule Limit As X Goes To 0 E X 1 X X 2
0 Response to " "
Post a Comment