JEE Main General Knowledge — practice by topic

To find the limit $$\\lim_{x \to 2} \\frac{x^2-4}{x-2}$$, follow these steps: 1. **Check for Direct Substitution:** Substitute $x = 2$ into the expression: $$\\frac{2^2 - 4}{2 - 2} = \\frac{4 - 4}{0} = \\frac{0}{0}$$ This is an **indeterminate form**, which means we must simplify the expression to find the limit. 2. **Factor the Numerator:** The numerator $x^2 - 4$ is a difference of squares ($a^2 - b^2 = (a-b)(a+b)$): $$x^2 - 4 = (x - 2)(x + 2)$$ 3. **Simplify the Fraction:** Substitute the factored form back into the limit: $$\\lim_{x \to 2} \\frac{(x - 2)(x + 2)}{x - 2}$$ As long as $x \\neq 2$, we can cancel the common term $(x - 2)$: $$\\lim_{x \to 2} (x + 2)$$ 4. **Evaluate the Limit:** Now, substitute $x = 2$ into the simplified expression: $$2 + 2 = 4$$ Therefore, the limit is **4**.

For a limit to exist at a specific point $x = c$, the function must approach the same finite value from both directions. 1. **The Left-Hand Limit**: This is the value the function approaches as $x$ gets closer to $c$ from values smaller than $c$. It is denoted as $\lim_{x \to c^-} f(x)$. 2. **The Right-Hand Limit**: This is the value the function approaches as $x$ gets closer to $c$ from values larger than $c$. It is denoted as $\lim_{x \to c^+} f(x)$. **The Formal Condition:** A limit $\lim_{x \to c} f(x)$ exists and equals $L$ if and only if: $$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$$ It is important to note that $f(c)$ does not need to be defined for the limit to exist. A function can have a 'hole' at $x=c$ (a removable discontinuity), and the limit will still exist as long as the paths from the left and right meet at that hole.

To find the derivative of $x^3$, we use the **Power Rule** of differentiation. The Power Rule states: $$\frac{d}{dx} x^n = nx^{n-1}$$ In this problem, our function is $x^3$, which means $n = 3$. 1. **Identify the exponent**: $n = 3$. 2. **Bring the exponent down** to the front as a multiplier: $3x^{...}$. 3. **Subtract 1 from the exponent**: $3 - 1 = 2$. Putting it all together: $$\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$$ Therefore, the correct answer is **3x²**.