Limits and Standard Limits: Quick Revision

A brief note on Limits and Standard Limits. Read it before a test. Try each question at the end before you open its answer.

The idea

• Limit: $\displaystyle\lim_{x \to a} f(x) = L$ means that $f(x)$ gets as close to $L$ as we like when $x$ is close to $a$, with $x \ne a$.
• The value $f(a)$ is not used. A function with a hole can still have a limit.
• Exists: the left-hand limit and the right-hand limit are both there and equal.
• No limit: a jump, growth to $+\infty$ on one side and $-\infty$ on the other, or swinging like $\sin\dfrac{1}{x}$.

Finding limits

• Polynomials: the limit at $a$ is the value at $a$.
• $\dfrac{0}{0}$: factorise and cancel, or multiply by the conjugate, then substitute.
• At infinity: divide the top and the bottom by the highest power of $x$.
• Degrees: lower on top gives $0$, equal gives the ratio of the leading coefficients, higher on top gives no finite limit.
• $\infty - \infty$ with roots: rationalise first.

The standard limits

• Sine and tangent: $\dfrac{\sin x}{x} \to 1$ and $\dfrac{\tan x}{x} \to 1$ as $x \to 0$, with $x$ in radians.
• Cosine: $\dfrac{1 - \cos x}{x^2} \to \dfrac{1}{2}$ as $x \to 0$.
• Powers: $\dfrac{x^n - a^n}{x - a} \to na^{n-1}$ as $x \to a$.
• Exponential and logarithm: $\dfrac{e^x - 1}{x} \to 1$, $\dfrac{\ln(1 + x)}{x} \to 1$ and $\dfrac{a^x - 1}{x} \to \ln a$ as $x \to 0$.
• Method: multiply and divide so that the inside of the function matches the denominator.

A worked example

Find $\displaystyle\lim_{x \to 0} \dfrac{\sin 5x}{3x}$.

1. Match the inside. $\dfrac{\sin 5x}{3x} = \dfrac{5}{3} \cdot \dfrac{\sin 5x}{5x}$.
2. Use the standard limit. The last fraction tends to $1$.
3. Answer. The limit is $\dfrac{5}{3}$.

Practice with new numbers

A program to try

The program looks at values on both sides of the point and compares them with the expected limit.

Mistakes to avoid

• Using $f(a)$ instead of the values near $a$.
• Looking at one side only. Check both.
• Cancelling terms, not factors.
• Degrees in $\dfrac{\sin x}{x}$. The limit $1$ needs radians.
• Treating $\infty - \infty$ as $0$. Rationalise.

Check yourself

1. Find $\displaystyle\lim_{x \to 4} \dfrac{x^2 - 16}{x - 4}$.

2. Find $\displaystyle\lim_{x \to \infty} \dfrac{6x^2 + 1}{3x^2 - x}$.

3. Find $\displaystyle\lim_{x \to 0} \dfrac{\sqrt{16 + x} - 4}{x}$.

4. Find $\displaystyle\lim_{x \to 0} \dfrac{\tan 4x}{x}$.

5. Find $\displaystyle\lim_{x \to 3} \dfrac{x^3 - 27}{x - 3}$.

Sources

• OpenStax, Calculus Volume 1, Chapter 2 on limits.
• NCERT, Mathematics Textbook for Class XI, Chapter 12.