1.3 Q-2

Question Statement

Evaluate the following limits using algebraic techniques:

1. $\lim _{x \rightarrow -1} \frac{x^{3}-x}{x+1}$
2. $\lim _{x \rightarrow 1} \frac{3x^{3}+4x}{x^{2}+x}$
3. $\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{2}-x-6}$
4. $\lim _{x \rightarrow 1} \frac{x^{3}-3x^{2}+3x-1}{x^{3}-x}$
5. $\lim _{x \rightarrow -1} \frac{x^{3}+x^{2}}{x^{2}-1}$
6. $\lim _{x \rightarrow 4} \frac{2x^{2}-32}{x^{3}-4x^{2}}$
7. $\lim _{x \rightarrow 2} \frac{\sqrt{x}-\sqrt{2}}{x-2}$
8. $\lim _{x \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$
9. $\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x^{m}-a^{m}}$

Background and Explanation

To solve limits algebraically, we use factorization, rationalization, and other algebraic techniques. These methods help to simplify expressions that might initially give indeterminate forms like $\frac{0}{0}$. It’s important to recognize when such techniques are necessary and to apply them systematically.

Solution

i. $\lim _{x \rightarrow -1} \frac{x^{3}-x}{x+1}$

1. Factor the numerator: $x^{3}-x = x(x^2 - 1) = x(x-1)(x+1)$

2. Simplify the expression: $\frac{x(x-1)(x+1)}{x+1}$

3. Cancel the $(x+1)$ terms: $\lim _{x \rightarrow -1} x(x-1)$

4. Substitute $x = -1$: $(-1)(-1-1) = 2$

Result: The limit is $2$.

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ii. $\lim _{x \rightarrow 1} \frac{3x^{3}+4x}{x^{2}+x}$

1. Substitute $x = 1$ directly: $\frac{3(1)^{3}+4(1)}{(1)^{2}+(1)} = \frac{3+4}{1+1} = \frac{7}{2}$

Result: The limit is $\frac{7}{2}$.

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iii. $\lim _{x \rightarrow 2} \frac{x^{3}-8}{x^{2}-x-6}$

1. Factor the numerator and denominator: $x^{3}-8 = (x-2)(x^2+2x+4)$ $x^{2}-x-6 = (x-3)(x+2)$

2. Simplify the expression: $\frac{(x-2)(x^2+2x+4)}{(x-3)(x+2)}$

3. Substitute $x = 2$: $\frac{(2-2)(2^2+2(2)+4)}{(2-3)(2+2)} = 0$

Result: The limit is $0$.

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iv. $\lim _{x \rightarrow 1} \frac{x^{3}-3x^{2}+3x-1}{x^{3}-x}$

1. Factor the numerator: $x^{3}-3x^{2}+3x-1 = (x-1)(x^2+x+1)$

2. Factor the denominator: $x^{3}-x = x(x^2-1) = x(x-1)(x+1)$

3. Cancel the $(x-1)$ terms: $\frac{x^2+x+1}{x(x+1)}$

4. Substitute $x = 1$: $\frac{(1)^2 + 1 + 1}{(1)(1+1)} = \frac{3}{2}$

Result: The limit is $\frac{3}{2}$.

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v. $\lim _{x \rightarrow -1} \frac{x^{3}+x^{2}}{x^{2}-1}$

1. Factor the numerator and denominator: $x^{3}+x^{2} = x^2(x+1)$ $x^{2}-1 = (x+1)(x-1)$

2. Simplify the expression: $\frac{x^2(x+1)}{(x+1)(x-1)}$

3. Cancel the $(x+1)$ terms: $\frac{x^2}{x-1}$

4. Substitute $x = -1$: $\frac{(-1)^2}{(-1-1)} = \frac{1}{-2} = -\frac{1}{2}$

Result: The limit is $-\frac{1}{2}$.

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vi. $\lim _{x \rightarrow 4} \frac{2x^{2}-32}{x^{3}-4x^{2}}$

1. Factor the numerator and denominator: $2x^{2}-32 = 2(x-4)(x+4)$ $x^{3}-4x^{2} = x^2(x-4)$

2. Simplify the expression: $\frac{2(x-4)(x+4)}{x^2(x-4)}$

3. Cancel the $(x-4)$ terms: $\frac{2(x+4)}{x^2}$

4. Substitute $x = 4$: $\frac{2(4+4)}{4^2} = \frac{16}{16} = 1$

Result: The limit is $1$ .

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vii. $\lim _{x \rightarrow 2} \frac{\sqrt{x}-\sqrt{2}}{x-2}$

1. Rationalize the numerator: $\frac{\sqrt{x}-\sqrt{2}}{x-2} \cdot \frac{\sqrt{x}+\sqrt{2}}{\sqrt{x}+\sqrt{2}}$

2. Simplify the numerator: $\frac{x-2}{(x-2)(\sqrt{x}+\sqrt{2})}$

3. Cancel the $(x-2)$ terms: $\frac{1}{\sqrt{x} + \sqrt{2}}$

4. Substitute $x = 2$: $\frac{1}{\sqrt{2} + \sqrt{2}} = \frac{1}{2\sqrt{2}}$

Result: The limit is $\frac{1}{2\sqrt{2}}$.

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viii. $\lim _{x \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h}$

1. Multiply numerator and denominator by the conjugate of the numerator: $\frac{\sqrt{x+h}-\sqrt{x}}{h} \cdot \frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$

2. Simplify the numerator: $\frac{h}{h(\sqrt{x+h}+\sqrt{x})}$

3. Cancel the $h$ terms: $\frac{1}{\sqrt{x+h} + \sqrt{x}}$

4. Substitute $x = 0$: $\frac{1}{\sqrt{h}}$

Result: The limit is $\frac{1}{\sqrt{h}}$.

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ix. $\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x^{m}-a^{m}}$

1. Use the binomial series approximation for $(x = a + h)$ and expand: $\frac{a^{n}\left[\left(1+\frac{h}{a}\right)^{n}-1\right]}{a^{m}\left[\left(1+\frac{h}{a}\right)^{m}-1\right]}$

2. Simplify the expression: $\frac{a^{n} \cdot \frac{n}{a}}{a^{m} \cdot \frac{m}{a}}$

3. Final result: $\frac{n}{m} \cdot a^{n-m}$

Result: The limit is $\frac{n}{m} \cdot a^{n-m}$.

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Key Formulas or Methods Used

• Factorization of polynomials
• Rationalization of expressions involving square roots
• Binomial series expansion

Summary of Steps

1. Simplify the expressions by factoring or rationalizing.
2. Substitute values where possible.
3. Apply standard algebraic techniques like canceling terms and using known limit laws.

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