Math Doubts

Evaluate $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x^2-3x+2}{x^2-5x+4}}$

It is purely an algebraic function in fraction form. It seems the limit of this algebraic function can be evaluated by the direct substitution method.

Evaluate Limit by Direct substitution

$\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x^2-3x+2}{x^2-5x+4}}$

Now, find the limit of the algebraic function as $x$ approaches $1$ by the direct substitution method.

$= \,\,\,$ $\dfrac{{(1)}^2-3{(1)}+2}{{(1)}^2-5{(1)}+4}$

$= \,\,\,$ $\dfrac{1-3+2}{1-5+4}$

$= \,\,\,$ $\dfrac{1+2-3}{1+4-5}$

$= \,\,\,$ $\dfrac{3-3}{5-5}$

$= \,\,\,$ $\dfrac{0}{0}$

The limit of the function as $x$ approaches $1$ is indeterminate. So, it can’t be evaluated by the direct substitution method and try another method.

Evaluate Limit by the factoring

$\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x^2-3x+2}{x^2-5x+4}}$

Each algebraic function in both numerator and denominator is a quadratic expression and each expression can be factored by the factorization method.

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x^2-2x-x+2}{x^2-4x-x+4}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x(x-2)-1(x-2)}{x(x-4)-1(x-4)}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{{(x-2)}{(x-1)}}{{(x-4)}{(x-1)}}}$

The factor $x-1$ is a common factor in both numerator and denominator, and they both get cancelled mathematically.

$= \,\,\,$ $\require{cancel} \displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{{(x-2)}\cancel{(x-1)}}{{(x-4)}\cancel{(x-1)}}}$

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x-2}{x-4}}$

Evaluate Limit by Direct Substitution

$= \,\,\,$ $\displaystyle \large \lim_{x \,\to\, 1}{\normalsize \dfrac{x-2}{x-4}}$

Now, the limit of the algebraic function can be evaluated by trying direct substitution method one more time.

$= \,\,\, \dfrac{1-2}{1-4}$

$= \,\,\, \dfrac{-1}{-3}$

$= \,\,\, \dfrac{1}{3}$

Math Doubts

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

Maths Topics

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

Maths Problems

A math help place with list of solved problems with answers 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