Question

evaluate Lim↓x-1(x^15-1/x^10-1)​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Limits

We are asked to evaluate the following limits:

$\longrightarrow \lim \limits_{x \to 1} \dfrac{x^{15} - 1}{x^{10} - 1}$

By directly substituting the limits x = 1, we get:

$\dfrac{1^{15} - 1}{1^{10} - 1} = \dfrac{1 - 1}{1 - 1} = \boxed{\dfrac{0}{0}}$

Which is an indeterminate form therefore we have to use another method to solve the problem.

Solution:

We can see that the factorisation is not possible. So, dividing numerator and denominator by $x - 1$.

$\implies \lim \limits_{x \to 1} \dfrac{\frac{x^{15} - 1}{x - 1}}{\frac{x^{10} - 1}{x - 1}}$

$\implies \lim \limits_{x \to 1} \Bigg\{\bigg(\dfrac{x^{15} - 1}{x - 1}\bigg) \div \bigg(\dfrac{x^{10} - 1}{x - 1}\bigg)\Bigg\}$

$\implies \lim \limits_{x \to 1} \bigg(\dfrac{x^{15} - 1^{10}}{x - 1}\bigg) \div \lim \limits_{x \to 1} \bigg(\dfrac{x^{10} - 1^{10}}{x - 1}\bigg)$

Now both the limits are in the form of $\lim \limits_{x \to a} \dfrac{x^n - a^n}{x - a} = na^{n-1}$ so applying this formula in it, we get:

$\implies \dfrac{15(1)^{15 - 1}}{10(1)^{10 - 1}}$

$\implies \dfrac{15(1)^{14}}{10(1)^{9}}$

$\implies \dfrac{15}{10}$

$\implies \dfrac{3}{2}$

Therefore the required answer is:

$\boxed{\lim \limits_{x \to 1} \dfrac{x^{15} - 1}{x^{10} - 1} = \dfrac{3}{2}}$