Question - Limit Calculation of Rational Function
Solution:
The limit given in the image is\[ \lim_{{x \to 5}} \frac{x - 5}{x^2 - 25} \]We can start by factoring the denominator:\[ x^2 - 25 = (x - 5)(x + 5) \]The expression for the limit then becomes:\[ \lim_{{x \to 5}} \frac{x - 5}{(x - 5)(x + 5)} \]We can simplify the expression by canceling out the common factor of $$x - 5$$ in the numerator and the denominator:\[ \lim_{{x \to 5}} \frac{1}{x + 5} \]Now we can directly substitute $$x = 5$$ into the simplified expression, as there are no more discontinuities:\[ \frac{1}{5 + 5} = \frac{1}{10} \]So, the value of the limit is $$\frac{1}{10}$$.
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