Limits and L’Hôpital’s rule

Many limits give 0/0 or ∞/∞ when you substitute. Then factor out the highest power or use L’Hôpital’s rule.

Rules

Same degree in numerator and denominator, $x \to \infty$ : ratio of the leading coefficients
L’Hôpital for $\frac{0}{0}$ or $\frac{\infty}{\infty}$ : $lim \frac{f}{g} = lim \frac{f ^{'}}{g ^{'}}$
Differentiate numerator and denominator separately, not with the quotient rule
Known: $lim_{x \to 0} \frac{s i n x}{x} = 1$ , $lim_{x \to 0} \frac{e ^{x} - 1}{x} = 1$

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Limits & L’Hôpital

Task

Evaluate

x \to 0 lim \frac{sin ( 5 x )}{x}

Limit

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Evaluate $x \to 0 lim \frac{e ^{5 x} - 1}{x}$

Worked solution

1.Inserting gives $\frac{0}{0}$ : L’Hôpital’s rule, differentiate numerator and denominator separately.
2. $lim_{x \to 0} \frac{5 e ^{5 x}}{1} = 5$

Typical mistake: Using the quotient rule for L’Hôpital; numerator and denominator are differentiated separately.

Common questions

Only when substitution gives 0/0 or ∞/∞. For other forms rearrange first, otherwise the result is wrong.