Question:
If
\[
\lim_{x\to -a}\frac{x^7+a^7}{x+a}=7,
\]
then \(a=\)
If
\[
\lim_{x\to -a}\frac{x^7+a^7}{x+a}=7,
\]
then \(a=\)
Show Hint
For limits of the form
\[
\lim_{x\to c}\frac{x^n-c^n}{x-c},
\]
use the standard result
\[
\lim_{x\to c}\frac{x^n-c^n}{x-c}=nc^{n-1}.
\]
Updated On: Jun 26, 2026
- \(\pm 7\)
- \(\pm 6\)
- \(\pm 1\)
- \(\pm 2\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Observe the indeterminate form.
At \[ x=-a, \] the numerator becomes \[ (-a)^7+a^7=-a^7+a^7=0 \] and the denominator becomes \[ -a+a=0. \] So, the given limit is of the form \[ \frac{0}{0}. \]
Step 2: Factorize \(x^7+a^7\).
Since \[ x^7+a^7=x^7-(-a)^7, \] we can write \[ x^7+a^7=(x+a)\left(x^6-ax^5+a^2x^4-a^3x^3+a^4x^2-a^5x+a^6\right) \]
Step 3: Cancel the common factor.
\[ \frac{x^7+a^7}{x+a} = x^6-ax^5+a^2x^4-a^3x^3+a^4x^2-a^5x+a^6 \]
Step 4: Substitute \(x=-a\).
\[ \lim_{x\to -a}\frac{x^7+a^7}{x+a} = 7a^6 \] Given that the limit is \(7\), so \[ 7a^6=7 \] \[ a^6=1 \] For real \(a\), \[ a=\pm 1. \]
Step 5: Final conclusion.
Therefore, \[ \boxed{a=\pm 1} \]
At \[ x=-a, \] the numerator becomes \[ (-a)^7+a^7=-a^7+a^7=0 \] and the denominator becomes \[ -a+a=0. \] So, the given limit is of the form \[ \frac{0}{0}. \]
Step 2: Factorize \(x^7+a^7\).
Since \[ x^7+a^7=x^7-(-a)^7, \] we can write \[ x^7+a^7=(x+a)\left(x^6-ax^5+a^2x^4-a^3x^3+a^4x^2-a^5x+a^6\right) \]
Step 3: Cancel the common factor.
\[ \frac{x^7+a^7}{x+a} = x^6-ax^5+a^2x^4-a^3x^3+a^4x^2-a^5x+a^6 \]
Step 4: Substitute \(x=-a\).
\[ \lim_{x\to -a}\frac{x^7+a^7}{x+a} = 7a^6 \] Given that the limit is \(7\), so \[ 7a^6=7 \] \[ a^6=1 \] For real \(a\), \[ a=\pm 1. \]
Step 5: Final conclusion.
Therefore, \[ \boxed{a=\pm 1} \]
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