Question

evalute the following​

Answer 1

Question :

$\dagger\ \; \displaystyle \sf \red{\lim_{x \to 2} \dfrac{3x^2-x-10}{x^2-4}}$

Solution :

It leads to indeterminant form , when apply the limit directly , i.e.,  $\dfrac{0}{0}$

$\displaystyle \sf \lim_{x \to 2} \dfrac{3x^2-x-10}{x^2-4}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{3x^2-6x+5x-10}{x^2-4}$

$\bullet\ \; \sf \orange{A^2-B^2=(A-B)(A+B)}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{3x(x-2)+5(x-2)}{(x-2)(x+2)}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{\cancel{(x-2)}(3x+5)}{\cancel{(x-2)}(x+2)}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{(3x+5)}{(x+2)}$

Apply limit ,

$\longrightarrow \displaystyle \sf \dfrac{(3(2)+5)}{((2)+2)}$

$\longrightarrow \displaystyle \sf \dfrac{6+5}{4}$

$\longrightarrow\ \; \displaystyle \sf \pink{\dfrac{11}{4}}$

★ ════════════════════ ★

$\bullet\ \; \displaystyle \sf \blue{\lim_{x \to 2} \dfrac{3x^2-x-10}{x^2-4} = \dfrac{11}{4}}$

Answer 2

Question :

$\dagger\ \; \displaystyle \sf \red{\lim_{x \to 2} \dfrac{3x^2-x-10}{x^2-4}}$

Solution :

It leads to indeterminant form , when apply the limit directly , i.e.,  $\dfrac{0}{0}$

$\displaystyle \sf \lim_{x \to 2} \dfrac{3x^2-x-10}{x^2-4}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{3x^2-6x+5x-10}{x^2-4}$

$\bullet\ \; \sf \orange{A^2-B^2=(A-B)(A+B)}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{3x(x-2)+5(x-2)}{(x-2)(x+2)}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{\cancel{(x-2)}(3x+5)}{\cancel{(x-2)}(x+2)}$

$\longrightarrow \displaystyle \sf \lim_{x \to 2} \dfrac{(3x+5)}{(x+2)}$

Apply limit ,

$\longrightarrow \displaystyle \sf \dfrac{(3(2)+5)}{((2)+2)}$

$\longrightarrow \displaystyle \sf \dfrac{6+5}{4}$

$\longrightarrow\ \; \displaystyle \sf \pink{\dfrac{11}{4}}$

★ ════════════════════ ★

$\bullet\ \; \displaystyle \sf \blue{\lim_{x \to 2} \dfrac{3x^2-x-10}{x^2-4} = \dfrac{11}{4}}$