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Model Paper 2 — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSModel Paper 21 Mark
MCQ
$\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}$ is equal to:
  • A
    $n a^{ n -1}$
  • B
    1
  • C
    $na^n$
  • D
    na
✓

Answer

(a) $n a^{n-1}$
Explanation: $\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}$
$\begin{array}{l}=\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}\left[\because f(x) \text { exists, } \lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a^{+}} f(x)\right] \\ =\lim _{h \rightarrow 0} \frac{(a+h)^n-a^n}{a+h-a} \\ =\lim _{h \rightarrow 0} a^n \frac{\left[\left(1+\frac{A}{a}\right)^n-1\right]}{h} \\ =a^{ n } \lim _{h \rightarrow 0}\left[1+ n \cdot \frac{h}{a}+\frac{n(n-1)}{2!} \frac{h^2}{a^2} \ldots+\ldots-1\right] \\ =a^{ n } \lim _{h \rightarrow 0}\left[\frac{n}{a}+\frac{h(h-1)}{2!} \frac{h}{a^2}+\ldots\right] \\ =a^{ n } \frac{n}{a} \\ =n a^{ n -1}\end{array}$

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