Memorize and apply the two fundamental special trigonometric limits that are crucial for differential calculus. Learn to recognize and solve variations of these limits through algebraic manipulation.
What is the first fundamental special trigonometric limit?
$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$
What is the second fundamental special trigonometric limit?
$$\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$$
Do the special trigonometric limits apply if \(x\) approaches a value other than \(0\), for example \(\lim_{x \to \pi} \frac{\sin(x)}{x}\)?
No. The special rules are valid only as \(x \to 0\). For any other value, you should try direct substitution first. For instance, $$\lim_{x \to \pi} \frac{\sin(x)}{x} = \frac{\sin(\pi)}{\pi} = \frac{0}{\pi} = 0$$
Evaluate: \(\lim_{x \to 0} \frac{x}{\sin(x)}\)
This can be rewritten as \(\lim_{x \to 0} \frac{1}{\frac{\sin(x)}{x}}\). Using the limit laws, this is \(\frac{\lim_{x \to 0} 1}{\lim_{x \to 0} \frac{\sin(x)}{x}} = \frac{1}{1} = 1\).
Evaluate: \(\lim_{x \to 0} \frac{\sin(4x)}{x}\)
To match the rule, we need the denominator to be \(4x\). Rewrite as $$\lim_{x \to 0} 4 \cdot \frac{\sin(4x)}{4x}$$ Let \(u=4x\). As \(x \to 0\), \(u \to 0\). The limit becomes $$4 \cdot \lim_{u \to 0} \frac{\sin(u)}{u} = 4 \cdot 1 = 4$$
What is the general result for \(\lim_{x \to 0} \frac{\sin(ax)}{bx}\), where \(a\) and \(b\) are constants?
The limit is \(\frac{a}{b}\). This is a useful shortcut derived from manipulating the expression to \(\frac{a}{b} \cdot \frac{\sin(ax)}{ax}\).
Evaluate: \(\lim_{x \to 0} \frac{\tan(x)}{x}\)
First, rewrite \(\tan(x)\) as \(\frac{\sin(x)}{\cos(x)}\). The limit becomes $$\lim_{x \to 0} \frac{\sin(x)}{x \cos(x)}$$ Split this into $$\lim_{x \to 0} \frac{\sin(x)}{x} \cdot \lim_{x \to 0} \frac{1}{\cos(x)} = 1 \cdot \frac{1}{1} = 1$$
This is a variation of the second special limit. Factor out a \(-1\) from the numerator: \(\lim_{x \to 0} -1 \cdot \frac{1 - \cos(x)}{x}\). This equals \(-1 \cdot 0 = 0\).
Evaluate: \(\lim_{x \to 0} \frac{\sin^2(x)}{x}\)
Rewrite the expression as \(\lim_{x \to 0} \sin(x) \cdot \frac{\sin(x)}{x}\). Split the limit: $$(\lim_{x \to 0} \sin(x)) \cdot (\lim_{x \to 0} \frac{\sin(x)}{x}) = 0 \cdot 1 = 0$$
Divide the numerator and denominator by \(x\): \(\lim_{x \to 0} \frac{\frac{\sin(5x)}{x}}{\frac{\sin(3x)}{x}}\). Using the rule \(\lim \frac{\sin(ax)}{x} = a\), this becomes \(\frac{5}{3}\).
