Question

evaluate following limit​

Answer 1

To Evaluate:-

$\displaystyle \lim_{y \to \ - 3} \frac{y {}^{5} + 243}{y {}^{3} + 27 }$

Solⁿ:-

$\implies\displaystyle \lim_{y \to \ - 3} \frac{(y {}^{5} + 243)/(y + 3)} {(y {}^{3} + 27) /(y + 3)}$

$\implies \frac{\displaystyle \lim_{y \to \ - 3} \: \: ( \frac{y {}^{5} - ( - 3) {}^{5} }{y - ( - 3)} )}{\displaystyle \lim_{y \to \ - 3 \: \:\: \frac{y {}^{3 \: - ( - 3) {}^{3} } }{y - ( - 3)} }}$

$\implies \: \frac{5 \: ( - 3) {}^{4} }{3 \: ( - 3) {}^{2} }$

$\implies \frac{5}{3} \times 9 = 15$

★ Formula used:-

$\displaystyle \lim_{x \to a} \frac{x {}^{n} - a {}^{n} }{x - a } = n.a {}^{n - 1}$

Therefore answer required is 15 ( Formula sab Yaad hai na :')?

Answer 2

Question :

$\dagger\ \; \displaystyle \sf \red{\lim_{y \to -3}\left[ \dfrac{y^5+243}{y^3+25}\right]}$

Solution :

$\displaystyle \sf \lim_{y \to -3}\left[ \dfrac{y^5+243}{y^3+25}\right]$

It leads to indeterminant form  $\dfrac{0}{0}$ , when we sub. the limits directly in the eq. . So ,

Apply L-Hopital's rule ,

$\longrightarrow \displaystyle \sf \lim_{y \to -3}\left[ \dfrac{\frac{d}{dx}(y^5+243)}{\frac{d}{dx}(y^3+25)}\right]$

$\longrightarrow \displaystyle \sf \lim_{y \to -3}\left[ \dfrac{5y^4}{3y^2}\right]$

Apply limit ,

$\longrightarrow \displaystyle \sf \left[ \dfrac{5(-3)^4}{3(-3)^2}\right]$

$\longrightarrow\ \; \displaystyle \sf \dfrac{5(81)}{3(9)}$

$\longrightarrow \displaystyle \sf\ \; \dfrac{5(9)}{3}$

$\longrightarrow\ \; \displaystyle \sf 5(3)$

$\longrightarrow \displaystyle \sf\ \; \pink{15}$

★ ═════════════════════ ★

$\bullet\ \; \displaystyle \sf \purple{\lim_{y \to -3}\left[ \dfrac{y^5+243}{y^3+25}\right]=15}$