Evaluate each limit using algebraic techniques. Use ( infty,-infty ) or ( D N E ) where appropriate. (a) ( lim _{x ightarrow 0} frac{x^{2}-25}{x^{2}-4 x-5} ) (b) ( lim _{x ightarrow 5} frac{x^{2}-25}{x^{2}-4 x-5} ) (c) ( lim _{x ightarrow 1} frac{7 x^{2}-4 x-3}{3 x^{2}-4 x+1} ) (d) ( lim _{x ightarrow-2} frac{x^{4}+5 x^{3}+6 x^{2}}{x^{2}(x+1)-4(x+1)} ) (e) ( lim _{x ightarrow-3}|x+1|+frac{3}{x} ) (f) ( lim _{x ightarrow 3} frac{sqrt{x+1}-2}{x^{2}-9} ) (g) ( lim _{x ightarrow 3} frac{sqrt{x^{2}+7}-3}{x+3} ) (h) ( lim _{x ightarrow 2} frac{x^{2}+2 x-8}{sqrt{x^{2}+5}-(x+1)} ) (i) ( lim _{y ightarrow 5}left(frac{2 y^{2}+2 y+4}{6 y-3} ight)^{1 / 3} )
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(a) \( \lim _{x \rightarrow 0} \frac{x^{2}-25}{x^{2}-4 x-5} \) We can factor the numerator and denominator: \( \lim _{x \rightarrow 0} \frac{(x-5)(x+5)}{(x-5)(x+1)} \) Now, we can cancel out the (x-5) terms: \( \lim _{x \rightarrow 0} \frac{x+5}{x+1} \) Now, we Show more…
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