Question

for the function ( f(x)=\frac{1}{x + 1} ), which of these could be a value of ( f(x) ) when ( x ) is close to -1?
oa - 1
ob. - 0.01
oc. 0.01
od. - 10,000

Explanation:

Step1: Analyze the function near \(x = - 1\)

When \(x\) approaches \(-1\) from the left (\(x\to - 1^{-}\)), \(x + 1\to0^{-}\), so \(F(x)=\frac{1}{x + 1}\to-\infty\). When \(x\) approaches \(-1\) from the right (\(x\to - 1^{+}\)), \(x + 1\to0^{+}\), so \(F(x)=\frac{1}{x + 1}\to+\infty\).

Step2: Check each option

• Option A: When \(x=-2\), \(F(-2)=\frac{1}{-2 + 1}=-1\), but this is not when \(x\) is close to \(-1\).
• Option B: \(-0.01\) is a small negative number. If \(F(x)=-0.01=\frac{1}{x + 1}\), then \(x+1=-100\), \(x=-101\) which is not close to \(-1\).
• Option C: \(0.01\) is a small positive number. If \(F(x)=0.01=\frac{1}{x + 1}\), then \(x + 1 = 100\), \(x=99\) which is not close to \(-1\).
• Option D: If \(F(x)=-10000=\frac{1}{x + 1}\), then \(x+1=-\frac{1}{10000}\), \(x=-1-\frac{1}{10000}\) which is very close to \(-1\) from the left - hand side.

Answer:

D. - 10,000