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Calculus Level 2

Consider the limit

lim ⁡ x → ∞ 7 x + 13 − 49 x 2 + 13 x \lim_{x\to \infty} 7x+13-\sqrt{49x^2+13x}

If the limit can be expressed as a b \frac{a}{b} , where gcd ⁡ ( a , b ) = 1 \gcd(a,b)=1 , enter a + b a+b . If the limit is + ∞ +\infty or − ∞ -\infty , enter -1 or -2, respectively. If the limit is an irrational real number or 0, enter 0.


The answer is 183.

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Kb E by kb e , 21, USA

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1 solution

Kb E
Sep 21, 2017

7 x + 13 − 49 x 2 + 13 x = 13 + 7 x − 49 x 2 + 13 x 1 = 13 + 49 x 2 − 49 x 2 − 13 x 7 x + 49 x 2 + 13 x = 13 + − 13 x 7 x + 49 x 2 + 13 = 13 + − 13 7 + 49 + 13 / x 2 7x+13-\sqrt{49x^2+13x} = 13 + \frac{7x-\sqrt{49x^2+13x}}{1} = 13+\frac{49x^2-49x^2-13x}{7x+\sqrt{49x^2+13x}} = 13+ \frac{-13x}{7x+\sqrt{49x^2+13}}= 13 + \frac{-13}{7+\sqrt{49+13/x^2}} lim ⁡ x → ∞ 13 + − 13 7 + 49 + 13 / x 2 = 13 − 13 14 = 169 14 ⟹ 169 + 14 = 183 \lim_{x\to\infty} 13+\frac{-13}{7+\sqrt{49+13/x^2}}= 13-\frac{13}{14}=\frac{169}{14} \implies 169+14=183

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