Question

evaluate the following limits. if needed, enter oo for \\( \infty \\) and -oo for \\( -\infty \\).
(a) \\( \lim _{x \
ightarrow \frac{11}{4}^{+}} \frac{31 x}{11-4 x}= \\)
(b) \\( \lim _{x \
ightarrow \frac{11}{4}^{-}} \frac{31 x}{11-4 x}= \\)
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Explanation:

Step1: Analyze the denominator when \(x\to\frac{11}{4}^+\)

When \(x\to\frac{11}{4}^+\), \(4x\to11^+\), so \(11 - 4x\to0^-\). And \(31x\to31\times\frac{11}{4}>0\). Then \(\frac{31x}{11 - 4x}\to-\infty\) since the numerator is positive and the denominator approaches \(0\) from the negative side.

Step2: Analyze the denominator when \(x\to\frac{11}{4}^-\)

When \(x\to\frac{11}{4}^-\), \(4x\to11^-\), so \(11 - 4x\to0^+\). And \(31x\to31\times\frac{11}{4}>0\). Then \(\frac{31x}{11 - 4x}\to+\infty\) since the numerator is positive and the denominator approaches \(0\) from the positive side.

Answer:

(a) \(-\infty\)
(b) \(\infty\)