1
Indeterminate Forms and L’Hospital’s Rule- HW Problems
Evaluate the following limits.
𝑥2 +2𝑥−3
1. lim
𝑥→1 𝑥−1
𝑥2 +2𝑥−3
2. lim
𝑥→∞ 2𝑥2 +1
ln(𝑥2 )
3. lim
𝑥→∞ 𝑥
cos(𝑥)−1
4. lim
𝑥→0 3𝑥2
tan−1 (2𝑥)
5. lim
𝑥→0 3𝑥
6. lim 𝑥 2 𝑒−𝑥
𝑥→∞
7. lim ([sin(𝑥 )][ln(𝑥 )])
𝑥→0+
1 1
8. lim+( − )
𝑥→0 𝑥 𝑒𝑥 −1
1
9. lim+(1 + tan(𝑥))𝑥
𝑥→0
10. lim (cot(𝑥 ) − csc(𝑥 ))
𝑥→0
11. lim+(sin(4𝑥))𝑥
𝑥→0
1 𝑥
12. lim ( )
𝑥→∞ 𝑥
9
(𝑥)
13. lim (𝑒𝑥 + 𝑥)
𝑥→∞
Indeterminate Forms and L’Hospital’s Rule- HW Problems
Evaluate the following limits.
𝑥2 +2𝑥−3
1. lim
𝑥→1 𝑥−1
𝑥2 +2𝑥−3
2. lim
𝑥→∞ 2𝑥2 +1
ln(𝑥2 )
3. lim
𝑥→∞ 𝑥
cos(𝑥)−1
4. lim
𝑥→0 3𝑥2
tan−1 (2𝑥)
5. lim
𝑥→0 3𝑥
6. lim 𝑥 2 𝑒−𝑥
𝑥→∞
7. lim ([sin(𝑥 )][ln(𝑥 )])
𝑥→0+
1 1
8. lim+( − )
𝑥→0 𝑥 𝑒𝑥 −1
1
9. lim+(1 + tan(𝑥))𝑥
𝑥→0
10. lim (cot(𝑥 ) − csc(𝑥 ))
𝑥→0
11. lim+(sin(4𝑥))𝑥
𝑥→0
1 𝑥
12. lim ( )
𝑥→∞ 𝑥
9
(𝑥)
13. lim (𝑒𝑥 + 𝑥)
𝑥→∞
