Math Doubts

Evaluate $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sqrt{1+x}-1}{x}}$ by Rationalization

The limit of square root of one plus $x$ minus one divided by $x$ is indeterminate as the value of $x$ approaches zero as per the direct substitution method.

$\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\sqrt{1+x}-1}{x}}$ $\,=\,$ $\dfrac{0}{0}$

limit question problem by rationalization

It is given in this limit question that the limit of square root of $1$ plus $x$ minus $1$ divided by $x$ should be evaluated by rationalization as the value of $x$ tends to $0$.

Remove the indeterminate form by Rationalisation

An expression in radical form the square root of $1$ plus $x$ is involved in forming the function in the numerator. So, let us try to remove the irrational form of the expression in the numerator by rationalizing it with its conjugate function.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \bigg(\dfrac{\sqrt{1+x}-1}{x}}$ $\times$ $1\bigg)$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \bigg(\dfrac{\sqrt{1+x}-1}{x}}$ $\times$ $\dfrac{\sqrt{1+x}+1}{\sqrt{1+x}+1}\bigg)$

Find the Product by simplifying Rational function

Now, it is time to multiply the two rational expressions consisting irrational form expressions by the multiplication rule of fractions.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\big(\sqrt{1+x}-1\big) \times \big(\sqrt{1+x}+1\big)}{x \times \big(\sqrt{1+x}+1\big)}}$

The product of the functions in the numerator can be multiplied by the difference of squares formula.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\big(\sqrt{1+x}\big)^2-1^2}{x\big(\sqrt{1+x}+1\big)}}$

Now, let us focus on simplifying the expressions in both numerator and denominator of the rational function.

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1+x-1}{x\big(\sqrt{1+x}+1\big)}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\cancel{1}+x-\cancel{1}}{x\big(\sqrt{1+x}+1\big)}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{x}{x\big(\sqrt{1+x}+1\big)}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{\cancel{x}}{\cancel{x}\big(\sqrt{1+x}+1\big)}}$

$=\,\,$ $\displaystyle \large \lim_{x\,\to\,0}{\normalsize \dfrac{1}{\sqrt{1+x}+1}}$

Evaluate the Limit by Direct substitution

Now, let us find the limit of the reciprocal of square root of one plus $x$ plus one by substituting $x$ is equal to zero directly.

$=\,\,$ $\dfrac{1}{\sqrt{1+0}+1}$

$=\,\,$ $\dfrac{1}{\sqrt{1}+1}$

$=\,\,$ $\dfrac{1}{1+1}$

$=\,\,$ $\dfrac{1}{2}$

Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved