Limits Practice Questions (Basics)

A limit is the value that a function approaches as the input gets closer to a particular value. It helps us understand the behavior of a function near a specific point.

Example: If a function gets closer and closer to 5 as x approaches 2, then the limit of the function as x approaches 2 is 5.

Solved Examples

Example 1: \lim_{x \to 6} \frac{x}{3}
Solution:

lim_x⇢6 x/3 = 6/3 = 2

Example 2: \lim_{x \to 2} \frac{x^2 - 4}{x - 2}
Solution:

As we know, (x² - 4) = (x² - 2²) = ( x - 2 )( x - 2 ) 

Now, lim_x⇢2 (x² - 4)/(x - 2)
= lim_x⇢2 (x- 2)(x + 2)/(x - 2)
= lim_x⇢2 (x + 2)
= 4

Example 3: \lim_{x \to \frac{1}{2}} \frac{2x - 1}{4x^2 - 1}
Solution:

As we know,

4x² − 1 = (2x)² − 1² = (2x+1) (2x−1)

Therefore,

\lim_{x\to\frac{1}{2}}\frac{2x-1}{4x^2-1} = \lim_{x\to\frac{1}{2}}\frac{2x-1}{(2x+1)(2x-1)}

=\lim_{x\to\frac{1}{2}}\frac{1}{2x+1}

=\frac{1}{2\left(\frac{1}{2}\right)+1}

=\frac{1}{2}

Example 4: Find the right-handed limit: \lim_{x \to 0^+} f(x) \quad \text{where} \quad f(x) = \begin{cases} 1 & \text{if } x > 0 \\0 & \text{if } x = 0 \\-1 & \text{if } x < 0\end{cases}
Solution:

Since we're asked to find the right-hand limit (x→0⁺), we are interested in the value of the function as x approaches 0 from the positive side (i.e., from values greater than 0).

• For x > 0, the function f(x) = 1
• Thus, the right-hand limit is:

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

Practice Questions

1. \lim_{x \to -2} \frac{x^2 - 1}{x + 2}
2. \lim_{x \to 3} \frac{x^3 - 27}{x - 3}
3. \lim_{x \to 5} \frac{x^2 + 2x - 15}{x - 5}
4. Find the right-handed limit: \lim_{x \to 2} \frac{x^2 - 4}{x - 2}