Question

lim x-(-1) x^25+1/x^15+1
$lim \\ x( - 1) \\ x^{25} + 1 \div x ^{15} + 1$

Answer 1

Calculus - Limits

We are asked to evaluate the following limits:

$\longrightarrow \lim \limits_{x \to -1} \dfrac{x^{25} + 1}{x^{15} + 1}$

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

$\dfrac{(-1)^{25} + 1}{(-1)^{15} + 1} = \dfrac{-1 + 1}{-1 + 1} = \boxed{\dfrac{0}{0}}$

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

Solution:

We can see that the factorisation is not possible. So, taking derivative of the numerator [top part of the fraction] and denominator [bottom part of the fraction], then evaluating the limits, we obtain:

$\implies \lim \limits_{x \to -1} \Bigg\{\dfrac{ \frac{d}{dx}(x^{25} + 1)}{ \frac{d}{dx}(x^{15} + 1)}\Bigg\}$

$\implies \lim \limits_{x \to -1} \Bigg\{\dfrac{25x^{25 - 1} + 0}{15x^{15 - 1} + 0}\Bigg\}$

$\implies \lim \limits_{x \to -1} \Bigg\{\dfrac{25x^{24} + 0}{15x^{14} + 0}\Bigg\}$

$\implies \lim \limits_{x \to -1} \Bigg\{\dfrac{25x^{24}}{15x^{14}}\Bigg\}$

Now we can substitute the limits, so by substituting the limits x → 1, we get the following results:

$\implies \dfrac{25( - 1)^{24}}{15( - 1)^{14} }$

$\implies \dfrac{25(1)}{15(1)}$

$\implies \dfrac{25}{15}$

$\implies \dfrac{5}{3}$

Therefore the required answer is:

$\boxed{\lim \limits_{x \to -1} \dfrac{x^{25} + 1}{x^{15} + 1} = \dfrac{5}{3}}$

$\rule{300}{2}$

Formula used:

The following are the formulas that have been used to find the solution:

$\boxed{\begin{array}{l}\bullet \;\;\dfrac{d}{dx}(x^n) = nx^{n-1} \\ \\ \bullet \;\;\dfrac{d}{dx}(a) = 0\end{array}}$