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

Expand Numerator: First, let's expand the numerator of the expression. (5+x)^2 - 3(5+x) - 10 = (25 + 10x + x^2) - (15 + 3x) - 10 = 25 + 10x + x^2 - 15 - 3x - 10 = x^2 + 7xSimplify Expression: Now, we can simplify the original expression by substituting the expanded numerator. lim_(x rarr0)((5+x)^(2)-3(5+x)-10)/(x) = lim_(x rarr0)(x^2 + 7x)/(x)Factor Out x: Next, we can factor out an x from the numerator. lim_(x rarr0)(x(x + 7))/(x)Cancel Out x: Since x is not equal to 0 (we are considering a limit as x approaches 0), we can cancel out the x in the numerator and the denominator. lim_(x rarr0)(x + 7)Substitute x=0: Now, we can directly substitute x = 0 into the simplified expression to find the limit. lim_(x rarr0)(x + 7) = 0 + 7 = 7

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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 $\lim _{x \rightarrow 0} \frac{(5+x)^{2}-3(5+x)-10}{x}$

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Algebra 2

Sum of finite series not start from 1

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Q. Find the limit of the expression

\lim _{x \rightarrow 0} \frac{(5+x)^{2}-3(5+x)-10}{x}

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