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Evaluate: $\displaystyle \lim_{x \to 1}\dfrac{{{x}^{15}}-1}{{{x}^{10}}-1}$

Last updated date: 23rd Jul 2024
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Hint: To evaluate the value of the given question, we will apply the limit and will check if we get the answer or not. If we do not get the answer, we will divide with a function $\left( x-1 \right)$ in the numerator and denominator of a given question to make calculation easy. After that we will get the limit function in the form of $\displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}$ and the limit function $\displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}$ is equal to $n{{a}^{n-1}}$. We will use this formula in the given question and will simplify to get the answer.

Complete step by step answer:
Since, the given limit function is:
$= \displaystyle \lim_{x \to 1}\dfrac{{{x}^{15}}-1}{{{x}^{10}}-1}$
Now, we will divide by $\left( x-1 \right)$ in the numerator and denominator of this limit function as:
$= \displaystyle \lim_{x \to 1}\dfrac{\left( \dfrac{{{x}^{15}}-1}{x-1} \right)}{\left( \dfrac{{{x}^{10}}-1}{x-1} \right)}$
We can write the above limit function as:
$= \dfrac{\displaystyle \lim_{x \to 1}\left( \dfrac{{{x}^{15}}-1}{x-1} \right)}{\displaystyle \lim_{x \to 1}\left( \dfrac{{{x}^{10}}-1}{x-1} \right)}$
Here, we get the limit function in the form of $\displaystyle \lim_{x \to a}\dfrac{{{x}^{n}}-{{a}^{n}}}{x-a}$ that is equal to $n{{a}^{n-1}}$. So, we will substitute $15$ for $n$ in the numerator and $10$ for $n$ in the denominator and will get $15{{\left( 1 \right)}^{15-1}}$ for numerator and $10{{\left( 1 \right)}^{10-1}}$ for denominator as:
$= \dfrac{15{{\left( 1 \right)}^{15-1}}}{10{{\left( 1 \right)}^{10-1}}}$
After doing required calculation in the above fraction as:
$= \dfrac{15{{\left( 1 \right)}^{14}}}{10{{\left( 1 \right)}^{9}}}$
As we know that any power of $1$ is always $1$. So, we will have from the above step as:
$= \dfrac{15\times \left( 1 \right)}{10\times \left( 1 \right)}$
The multiplication of any number with one always gives that number as:
$= \dfrac{15}{10}$
Now, we will simplify it into simplest form of fraction as:
$= \dfrac{3}{2}$
Hence, the value of limit function $\displaystyle \lim_{x \to 1}\dfrac{{{x}^{15}}-1}{{{x}^{10}}-1}$ is $\dfrac{3}{2}$.

Note: For any limit function, we can use some techniques to evaluate the value of it if the limit function provides its values in the form of $\dfrac{0}{0}$ after applying the limit. Here are some methods such as putting the value of the limit, factorization, rationalization, finding the least common denominator L-Hospital rule, etc.