Evaluate lim_(x rarr10)(x^(2)-99)/(x-10)

Identify Problem Type: Identify the type of problem. We are asked to find the limit of a rational function as x approaches a specific value. This is a limit problem in calculus.Substitute Value: Substitute the value of x into the function to see if the function is defined at that point. lim_(x -> 10)(x^2 - 99) / (x - 10) = (10^2 - 99) / (10 - 10) = (100 - 99) / 0 = 1 / 0 We encounter a division by zero, which means the function is not defined at x = 10, and we have an indeterminate form of type 0/0.Factor Numerator: Factor the numerator to simplify the expression. The numerator x^2 - 99 can be factored as (x + 9)(x - 10) because it is a difference of squares.Cancel Common Factors: Cancel out the common factors in the numerator and the denominator. The (x - 10) term in the numerator cancels with the (x - 10) term in the denominator, simplifying the expression to: lim_(x -> 10)(x + 9)Substitute Simplified Function: Substitute the value of x into the simplified function. Now that the indeterminate form has been resolved, we can substitute x = 10 directly into the simplified function: lim_(x -> 10)(x + 9) = 10 + 9 = 19

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Evaluate
lim_(x rarr10)(x^(2)-99)/(x-10)

Evaluate $\lim_{x \to 10}\frac{x^{2}-99}{x-10}$

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate

\lim_{x \to 10}\frac{x^{2}-99}{x-10}

Identify Problem Type: Identify the type of problem. $\newline$ We are asked to find the limit of a rational function as $x$ approaches a specific value. This is a limit problem in calculus.
Substitute Value: Substitute the value of $x$ into the function to see if the function is defined at that point. $\newline$ $\lim_{x \to 10}\frac{x^2 - 99}{x - 10} = \frac{10^2 - 99}{10 - 10} = \frac{100 - 99}{0} = \frac{1}{0}$ $\newline$ We encounter a division by zero, which means the function is not defined at $x = 10$ , and we have an indeterminate form of type $0/0$ .
Factor Numerator: Factor the numerator to simplify the expression. $\newline$ The numerator $x^2 - 99$ can be factored as $(x + 9)(x - 10)$ because it is a difference of squares.
Cancel Common Factors: Cancel out the common factors in the numerator and the denominator. $\newline$ The $(x - 10)$ term in the numerator cancels with the $(x - 10)$ term in the denominator, simplifying the expression to: $\newline$ $\lim_{x \to 10}(x + 9)$
Substitute Simplified Function: Substitute the value of $x$ into the simplified function. $\newline$ Now that the indeterminate form has been resolved, we can substitute $x = 10$ directly into the simplified function: $\newline$ $\lim_{x \to 10}(x + 9) = 10 + 9 = 19$