MH Board · Chapter

Limits

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Up nextLimits and one-sided limits $\lim_{x \to a} f(x)$ exists only when the left-hand and right-hand limits are equal; $f(a)$ itself doesn't matterStart

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The idea0 / 2

Limits and one-sided limits $\lim_{x \to a} f(x)$ exists only when the left-hand and right-hand limits are equal; $f(a)$

itself doesn't matter

Start

Algebra of limits and substitutionLimits of sums, products and quotients are sums, products and quotients of limits; if substituting gives a number, that's the limitLocked

0/0 forms0 / 3

Cancelling the common factor $\frac{0}{0}$ at $x = a$ means $(x - a)$ divides both: factorise, cancel, substituteLocked
RationalizingMultiply by the conjugate $\sqrt{a} + \sqrt{b}$
The limit of (xⁿ − aⁿ)/(x − a) $\lim_{x \to a}\frac{x^n - a^n}{x - a} = na^{n-1}$ ,

Special limits0 / 4

Limits at infinityDivide by the highest power of $x$ in the denominator; $\frac{1}{x^k} \to 0$ Locked
Trigonometric limits $\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = 1$
Exponential limits $\lim_{x \to 0}\frac{a^x - 1}{x} = \log a$
Logarithmic limits and e $\lim_{x \to 0}\frac{\log(1 + x)}{x} = 1$

to clear the root, then cancel

Locked

for any rational

n

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with

\theta

in radians; make the angle and the denominator match

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and

\lim_{x \to 0}\frac{e^x - 1}{x} = 1

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and

\lim_{x \to 0}(1 + x)^{\frac{1}{x}} = e

Locked