Question

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find \\(\lim\limits_{x\to 5} \dfrac{x + 2}{x^2 - 5}\\).
choose 1 answer:
a \\(-\dfrac{3}{25}\\)
b \\(\dfrac{7}{20}\\)
c \\(-\dfrac{7}{25}\\)
d the limit doesnt exist
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Explanation:

Step1: Check continuity at \( x = 5 \)

The function \( f(x)=\frac{x + 2}{x^{2}-5} \) is a rational function. A rational function is continuous everywhere except where the denominator is zero. Let's check the denominator at \( x = 5 \): \( 5^{2}-5=25 - 5=20
eq0 \). So the function is continuous at \( x = 5 \).

Step2: Use direct substitution

For a continuous function at a point \( a \), \( \lim_{x
ightarrow a}f(x)=f(a) \). So we substitute \( x = 5 \) into the function:
\( \lim_{x
ightarrow5}\frac{x + 2}{x^{2}-5}=\frac{5 + 2}{5^{2}-5}=\frac{7}{20} \)

Answer:

B. \( \frac{7}{20} \)