Question

If \[ \lim_{x\to -a}\frac{x^7+a^7}{x+a}=7, \] then \(a=\)

• A. \(\pm 7\)
• B. \(\pm 6\)
• C. \(\pm 1\)
• D. \(\pm 2\)

Hint:

For limits of the form \[ \lim_{x\to c}\frac{x^n-c^n}{x-c}, \] use the standard result \[ \lim_{x\to c}\frac{x^n-c^n}{x-c}=nc^{n-1}. \]

Answer 1

The Correct Option is C

Step 1: Use the standard limit formula.
\(\lim_{x \to c}\dfrac{x^n - c^n}{x-c} = n\,c^{n-1}\). Here, \(\dfrac{x^7+a^7}{x+a} = \dfrac{x^7-(-a)^7}{x-(-a)}\), so \(c = -a,\; n=7\).

Step 2: Apply the formula.
\[\lim = 7(-a)^6 = 7a^6 = 7\] \[a^6 = 1 \Rightarrow a = \pm 1\]
\[ \boxed{a = \pm 1} \]