Physics and Mathematics
Algebraic Limits By Using Standard Results Example 1
Practice Questions with Step-by-Step Solutions
Question 1. Evaluate:
[\lim_{x \to 3} \dfrac{x^{2} − 9}{x − 3}]
Step-by-Step Solution:
Step 1: Substitute [x = 3] to check the form
Numerator:
[3^{2} − 9 = 9 − 9 = 0]
Denominator:
[3 − 3 = 0]
So, the form is [0/0], which is indeterminate.
Step 2: Compare with the standard result
[\lim_{x \to a} \dfrac{x^{n} − a^{n}}{x − a} = n a^{n−1}]
Here:
[n = 2]
[a = 3]
Step 3: Apply the standard result
Value of the limit:
[2 × 3 = 6]
Final Answer:
[6]
Question 2. Evaluate:
[\lim_{x \to 2} \dfrac{x^{3} − 8}{x − 2}]
Step-by-Step Solution:
Step 1: Direct substitution
[2^{3} − 8 = 8 − 8 = 0]
[2 − 2 = 0]
So the form is [0/0].
Step 2: Identify parameters
[n = 3], [a = 2]
Step 3: Apply standard result
[3 × 2^{2} = 3 × 4 = 12]
Final Answer:
[12]
Question 3. Evaluate:
[\lim_{x \to 1} \dfrac{x^{4} − 1}{x − 1}]
Step-by-Step Solution:
Step 1: Substitute [x = 1]
[1^{4} − 1 = 0], [1 − 1 = 0]
Form = [0/0]
Step 2: Compare with standard form
Here [n = 4], [a = 1]
Step 3: Apply formula
[4 × 1^{3} = 4]
Final Answer:
[4]
Question 4. Evaluate:
[\lim_{x \to 5} \dfrac{x^{3} − 125}{x − 5}]
Step-by-Step Solution:
Step 1: Substitute [x = 5]
[125 − 125 = 0], [5 − 5 = 0]
Form = [0/0]
Step 2: Identify values
[n = 3], [a = 5]
Step 3: Apply standard result
[3 × 5^{2} = 3 × 25 = 75]
Final Answer:
[75]
Question 5. Evaluate:
[\lim_{x \to 4} \dfrac{x^{2} − 16}{x − 4}]
Step-by-Step Solution:
Step 1: Substitute [x = 4]
[16 − 16 = 0], [4 − 4 = 0]
Indeterminate form [0/0]
Step 2: Identify parameters
[n = 2], [a = 4]
Step 3: Apply formula
[2 × 4 = 8]
Final Answer:
[8]
Question 6. Evaluate:
[\lim_{x \to a} \dfrac{x^{4} − a^{4}}{x − a}]
Step-by-Step Solution:
Step 1: Substitute [x = a]
[a^{4} − a^{4} = 0], [a − a = 0]
Form = [0/0]
Step 2: Match with standard result
[n = 4]
Step 3: Apply formula
[4a^{3}]
Final Answer:
[4a^{3}]
Question 7. Evaluate:
[\lim_{x \to 1} \dfrac{x^{7} − 1}{x − 1}]
Step-by-Step Solution:
Step 1: Substitute [x = 1]
[1 − 1 = 0], [1 − 1 = 0]
Form = [0/0]
Step 2: Identify values
[n = 7], [a = 1]
Step 3: Apply standard result
[7 × 1^{6} = 7]
Final Answer:
[7]
Question 8. Evaluate:
[\lim_{x \to 2} \dfrac{x^{5} − 32}{x − 2}]
Step-by–Step Solution:
Step 1: Substitute [x = 2]
[32 − 32 = 0], [2 − 2 = 0]
Form = [0/0]
Step 2: Identify parameters
[n = 5], [a = 2]
Step 3: Apply standard formula
[5 × 2^{4} = 5 × 16 = 80]
Final Answer:
[80]
Question 9. Evaluate:
[\lim_{x \to 6} \dfrac{x^{2} − 36}{x − 6}]
Step-by-Step Solution:
Step 1: Substitute [x = 6]
[36 − 36 = 0], [6 − 6 = 0]
Indeterminate form [0/0]
Step 2: Identify values
[n = 2], [a = 6]
Step 3: Apply standard result
[2 × 6 = 12]
Final Answer:
[12]
Question 10. Evaluate:
[\lim_{x \to 0} \dfrac{x^{4} − 0}{x}]
Step-by-Step Solution:
Step 1: Simplify the expression
[\dfrac{x^{4}}{x} = x^{3}]
Step 2: Substitute [x = 0]
[0^{3} = 0]
Final Answer:
[0]
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