Question

How do you find the limit at a hole? The function I have is $\displaystyle \lim_{x \to 9}\dfrac{x-9}{\sqrt{x+7}-4}$ . How do I find the limit, if the function gives $\dfrac{0}{0}$ ?

Answer 1

Hint: Since, the given function contains a square root in the denominator and a polynomial expression. So, we will use the method of rationalization with respect to the denominator in which we will multiply with $\sqrt{x+7}+4$ in numerator and denominator. After simplifying it we will find a simple polynomial expression where we can apply the limit and will get the value of the limit.

Complete step by step solution:
Since, the given function in the question is:
$\Rightarrow \displaystyle \lim_{x \to 9}\dfrac{x-9}{\sqrt{x+7}-4}$
Since, denominator has the under root as $\sqrt{x+7}-4$ , we will take its rationalization that is $\sqrt{x+7}+4$ and will multiply with this rationalization in the numerator and denominator of the above function as:
$\Rightarrow \displaystyle \lim_{x \to 9}\dfrac{x-9}{\sqrt{x+7}-4}\times \dfrac{\sqrt{x+7}+4}{\sqrt{x+7}+4}$ … $\left( i \right)$
Here, we will apply the rule of FOIL in the denominator to solve the equation as:
$\Rightarrow \left( \sqrt{x+7}-4 \right)\times \left( \sqrt{x+7}+4 \right)$
So, according to the FOIL method to get the product of tow binomials we will add the multiplication of first term with first term, second term with first term , first term with second term and second term with second term. Here, we can understand it easily by example:
$\Rightarrow \left( a+b \right)\left( c+d \right)=ac+bc+ad+bd$
So, the multiplication of the denominator will be:
$\Rightarrow \sqrt{x+7}\times \sqrt{x+7}-4\times \sqrt{x+7}+4\times \sqrt{x+7}-4\times 4$
Here, two equal like terms will be cancel out. So, we will get:
$\Rightarrow \sqrt{x+7}\times \sqrt{x+7}-4\times 4$
After solving the above equation, we will have:
$\Rightarrow x+7-16$
$\Rightarrow x-9$
Since, we got the solution of the denominator; we will put this value in the equation $\left( i \right)$ as:
\[\Rightarrow \displaystyle \lim_{x \to 9}\dfrac{\left( x-9 \right)\left( \sqrt{x+7}+4 \right)}{x-9}\]
Here, we will again cancel out the equal like terms. So we will have the above equation as:
\[\Rightarrow \displaystyle \lim_{x \to 9}\sqrt{x+7}+4\]
Now, we can calculate the limit as:
\[\Rightarrow \sqrt{9+7}+4\]
Here, we will solve the above function using necessary calculation as:
\[\Rightarrow \sqrt{16}+4\]
Since, the $16$ is the square of $4$ . So, we can write the square root of $16$ is $4$ as:
\[\Rightarrow 4+4\]
And this will gives the final answer as:
\[\Rightarrow 8\]
Hence, the limit of the function $\displaystyle \lim_{x \to 9}\dfrac{x-9}{\sqrt{x+7}-4}$ is $8$ .

Note: When we have to get the limit of any function, we will use four techniques that are: Putting the value of the limit, factorization, rationalization and finding the least common denominator. First of all we will use the method for getting the limit by putting the value of the limit. Sometimes we will get the limit but sometimes we will get the limit in the form of $\dfrac{0}{0}$ . Then, we will use another method that is factorization. If we get the limit, there is no need to go further but we do not have the limit value, we will use the next method, the method of rationalization. Similarly, we get the result; we will stop the process or will use the next method that is finding the least common denominator method.