Math Doubts

Solve $\large \displaystyle \lim_{x \,\to\, 4} \dfrac{3-\sqrt{5+x}}{x-4}$

$x$ is a literal and its represents the value of limit of a function. The literal formed two functions and and the faction of them is required to find mathematical if the value of $x$ tends to $4$.

$\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{3-\sqrt{5+x}}{x-4}$

Step: 1

Firstly, test this algebraic function when $x$ tends to $4$.

$\implies \dfrac{3-\sqrt{5+4}}{x-4}$ $=$ $\dfrac{3-\sqrt{9}}{4-4}$

$= \dfrac{3-3}{4-4}$

$= \dfrac{0}{0}$

The algebraic function became indeterminate when the value of $x$ approaches to $0$. Hence, the limit problem should be simplified in alternative method.

Step: 2

The numerator is in radical form. So, multiply both numerator and denominator by its rationalizing factor.

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{3-\sqrt{5+x}}{x-4}$ $\times$ $\dfrac{3+\sqrt{5+x}}{3+\sqrt{5+x}}$

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{(3-\sqrt{5+x})(3+\sqrt{5+x})}{(x-4)(3+\sqrt{5+x})}$

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{{(3)}^2-{(\sqrt{5+x})}^2}{(x-4)(3+\sqrt{5+x})}$

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{9-(5+x)}{(x-4)(3+\sqrt{5+x})}$

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{9-5-x}{(x-4)(3+\sqrt{5+x})}$

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{4-x}{(x-4)(3+\sqrt{5+x})}$

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{-(x-4)}{(x-4)(3+\sqrt{5+x})}$

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\require{cancel} \dfrac{-\cancel{(x-4)}}{\cancel{(x-4)}(3+\sqrt{5+x})}$

Step: 3

Substitute $x = 4$ to determine the value of the algebraic function when $x$ tends to $4$.

$=$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{-1}{3+\sqrt{5+x}}$

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

$=$ $\dfrac{-1}{3+\sqrt{9}}$

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

$=$ $\dfrac{-1}{6}$

$\therefore \,\,\,\,\,\,$ $\large \displaystyle \lim_{x \,\to\, 4}$ $\dfrac{3-\sqrt{5+x}}{x-4}$ $=$ $-\dfrac{1}{6}$

Therefore, it is the required solution for this limit problem in mathematics.



Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more