Limit, Continuity and Differentiability Mock Test for JEE Main 2025-26 Preparation

To check the continuity of a function at a point x = a:

• The function, f(x), must be defined at x = a.
• The limit of f(x) as x approaches a must exist.
• The value of the function at x = a must equal the limit, i.e., lim_x→a f(x) = f(a).

Right-hand limit (RHL) refers to the limit of a function as the input approaches a specific point from the right side (x → a⁺), while the left-hand limit (LHL) is the limit as the input approaches from the left (x → a^-). For a function to be continuous at a point, both limits must exist and be equal to each other and the value of the function at that point.

A function is said to be differentiable at a point if its derivative exists at that point. In other words, if the function has a defined and finite slope (rate of change) at a specific value, it is differentiable there. Every differentiable function is continuous, but not every continuous function is differentiable.

No, every continuous function is not necessarily differentiable. While differentiability implies continuity, the reverse is not always true. For example, a function with a sharp corner or cusp (like |x| at x = 0) is continuous but not differentiable at that point.

Some common techniques to evaluate limits are:

• Direct substitution
• Factoring
• Rationalizing
• Using standard limits formulas
• L'Hospital's Rule (for indeterminate forms like 0/0 or ∞/∞)

An indeterminate form arises when the direct substitution in a limit leads to an undefined or ambiguous value, such as 0/0, ∞/∞, 0×∞, ∞-∞, 0⁰, and 1^∞. These require special techniques (like factoring or L'Hospital's Rule) to resolve.

For piecewise functions, check continuity and differentiability at points where the formula changes.

1. Ensure the function values and limits from both sides at the joining point are equal (continuity).
2. Check that the derivatives from left and right exist and are equal at that point (differentiability).

Limits, continuity, and differentiability form the foundation of calculus concepts, especially in defining and understanding derivatives and integrals. Mastering these concepts is essential for solving problems in mathematics, engineering, physics, and applied sciences.

Multiple choice tests on limits and continuity typically include:

• Evaluating limits using basic algebra or formulas
• Identifying points of discontinuity
• Selecting correct statements about differentiability
• Solving for unknowns given continuity conditions

To quickly check for discontinuity: look for abrupt jumps, holes, or undefined points in the function. For non-differentiability: check for sharp corners, cusps, or points where the left and right derivatives differ or do not exist.

You can find practice tests, mock tests, and PDF worksheets for limits, continuity, and differentiability on educational platforms like Vedantu, NCERT, and other exam preparation websites. These resources include multiple choice questions, solutions, and concept explanations aligned with the current syllabus.