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Find the limit of the expression lim_(x rarr0)((5+x)^(2)-3(5+x)-10)/(x)

Find the limit of the expression lim⁡x→0(5+x)2−3(5+x)−10x \lim _{x \rightarrow 0} \frac{(5+x)^{2}-3(5+x)-10}{x}

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Full solution

Q. Find the limit of the expression lim⁡x→0(5+x)2−3(5+x)−10x \lim _{x \rightarrow 0} \frac{(5+x)^{2}-3(5+x)-10}{x}
  1. Expand Numerator: First, let's expand the numerator of the expression.\newline(5+x)2−3(5+x)−10(5+x)^2 - 3(5+x) - 10\newline= (25+10x+x2)−(15+3x)−10(25 + 10x + x^2) - (15 + 3x) - 10\newline= 25+10x+x2−15−3x−1025 + 10x + x^2 - 15 - 3x - 10\newline= x2+7xx^2 + 7x
  2. Simplify Expression: Now, we can simplify the original expression by substituting the expanded numerator.\newlinelim⁡x→0(5+x)2−3(5+x)−10x\lim_{x \to 0}\frac{(5+x)^{2}-3(5+x)-10}{x}\newline= lim⁡x→0x2+7xx\lim_{x \to 0}\frac{x^2 + 7x}{x}
  3. Factor Out xx: Next, we can factor out an xx from the numerator.lim⁡x→0x(x+7)x\lim_{x \rightarrow 0}\frac{x(x + 7)}{x}
  4. Cancel Out xx: Since xx is not equal to 00 (we are considering a limit as xx approaches 00), we can cancel out the xx in the numerator and the denominator.\newlinelim⁡x→0(x+7)\lim_{x \to 0}(x + 7)
  5. Substitute x=0x=0: Now, we can directly substitute x=0x = 0 into the simplified expression to find the limit.lim⁡x→0(x+7)=0+7=7\lim_{x \to 0}(x + 7) = 0 + 7 = 7

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