Complete Function Limit Example Problems + Solutions

A collection of complete function limit example problems from basic to HOTS. Includes approach tables, graphs, and detailed solutions. Perfect for grade 11 high school students who want to master function limits thoroughly.

🎯 Learning Objectives:

• Solve algebraic and trigonometric function limit problems.
• Use numerical (table), graphical, and analytical methods.
• Identify indeterminate forms and use the appropriate solving techniques.

📊 Table Approach: Left and Right Hand Limits

Example 1:

Calculate:

$$ \lim_{x \to 2} f(x) = x^2 $$

Create an Approach Table:

👉 From the left and right, the value approaches 4, so:

$$ \lim_{x \to 2} x^2 = \boxed{4} $$

📈 Graphical Approach

Example 2:

$$ f(x) = \begin{cases} x^2 - 1, & x < 1 \
2x + 1, & x \geq 1 \end{cases} $$

Determine:

$$ \lim_{x \to 1} f(x) $$

Solution:

• From the left: $\lim_{x \to 1^-} = 1^2 - 1 = 0$
• From the right: $\lim_{x \to 1^+} = 2(1) + 1 = 3$

Since left ≠ right, then:

$$ \lim_{x \to 1} f(x) \text{ does not exist.} $$

📚 Basic Practice Problems

Problem 1:

$$ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} $$

Factorization:

$$ \frac{(x - 3)(x + 3)}{x - 3} = x + 3 \Rightarrow \boxed{6} $$

Problem 2:

$$ \lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x} $$

Multiply by the conjugate:

$$ \cdot \frac{\sqrt{1 + x} + 1}{\sqrt{1 + x} + 1} = \frac{x}{x(\sqrt{1 + x} + 1)} = \frac{1}{\sqrt{1 + x} + 1} \Rightarrow \boxed{\frac{1}{2}} $$

🔥 HOTS Problems

Problem 3:

$$ \lim_{x \to 0} \frac{\tan x - \sin x}{x^3} $$

Use Taylor expansion (advanced class) or L’Hôpital’s Rule up to 3 times.

Final result:

$$ \boxed{\frac{1}{2}} $$

Problem 4:

$$ f(x) = \begin{cases} \frac{x^2 - 4}{x - 2}, & x < 2 \
ax + b, & x \geq 2 \end{cases} $$

If the limit as $x \to 2$ exists, determine the values of $a$ and $b$.

Solution:

• From the left:

$$ \frac{x^2 - 4}{x - 2} = x + 2 \Rightarrow \lim_{x \to 2^-} = 4 $$

• From the right:

$$ \lim_{x \to 2^+} = 2a + b $$

For the limit to exist:

$$ 2a + b = 4 \Rightarrow \text{This is a system of equations: many solutions} $$

🧪 Independent Practice Problems

1. $\displaystyle \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$
2. $\displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x^2}$
3. $\displaystyle \lim_{x \to \infty} \frac{2x^2 + 3}{x^2 - 1}$
4. $\displaystyle \lim_{x \to 0} \frac{\ln(1 + x)}{x}$
5. If $f(x) = \begin{cases} k(x - 1)^2, & x < 1 \
   mx + 1, & x \geq 1 \end{cases}$, Determine $k$ and $m$ so that the limit as $x \to 1$ is continuous.

🔗 Other Articles You Should Learn:

• 👉 [Functions and Function Graphs]
• 👉 [Derivatives: Definition and Application]