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Statistics · Ch 8 — Limit

Standard Limits

3

Standard Limits

Rather than factorising or rationalising from scratch every time, certain limit forms recur so often — in this Gujarat Std-12 Statistics chapter and in the differentiation chapter that follows it — that they are memorised as standard limits and applied directly.

1. The power formula

lim⁡x→axn−anx−a=n a n−1,n a positive integer\lim_{x\to a}\dfrac{x^{n}-a^{n}}{x-a}=n\,a^{\,n-1}, \qquad n \text{ a positive integer}

This single formula instantly evaluates any limit of the 00\frac{0}{0} form built from a difference of like powers, without factorising by hand.

2. The exponential formula

lim⁡x→0ex−1x=1and more generallylim⁡x→0ax−1x=log⁡ea\lim_{x\to 0}\dfrac{e^{x}-1}{x}=1 \qquad\text{and more generally}\qquad \lim_{x\to 0}\dfrac{a^{x}-1}{x}=\log_{e}a

3. The logarithmic formula

lim⁡x→0log⁡e(1+x)x=1\lim_{x\to 0}\dfrac{\log_{e}(1+x)}{x}=1

4. The number ee itself, as a limit

lim⁡n→∞(1+1n)n=e≈2.71828\lim_{n\to \infty}\left(1+\dfrac{1}{n}\right)^{n}=e \approx 2.71828 …

Definition 1The Four Standard Limits

lim⁡x→axn−anx−a=nan−1\displaystyle\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}; lim⁡x→0ex−1x=1\displaystyle\lim_{x\to0}\frac{e^x-1}{x}=1; lim⁡x→0ax−1x=log⁡ea\displaystyle\lim_{x\to0}\frac{a^x-1}{x}=\log_ea; lim⁡x→0log⁡e(1+x)x=1\displaystyle\lim_{x\to0}\frac{\log_e(1+x)}{x}=1; $\displaystyle\lim_{n\ …

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