Sample Problem on Limit Formula
Question 1: Solve, limx⇢0 (x – sinx ) /(1 – cosx).
Solution:
Using L-hospital,
limx ⇢ 0 (1 – cosx) / (sinx)
limx ⇢ 0 sinx / cosx = sin(0) / cos(0) = 0/1 = 0
Question 2: Solve, limx ⇢ 0 (e2x -1) / sin4x.
Solution:
Using L-hospital
limx ⇢ 0 (2)(e2x) / cos4x
limx ⇢ 0 2(e0) / cos4(0) = 2/1= 2
Question 3: Solve, limx ⇢ 0 (1 – cosx) / x2
Solution:
Using L-hospital
limx ⇢ 0 sinx /2x = 1/2 {sinx/x = 1}
Question 4: Solve, limx ⇢ ∞
Solution:
limx ⇢ ∞ (1 +
\frac{sinx}{x} )1 + limx ⇢ ∞
\frac{sinx}{x} As we know, x = ∞
So 1/x = 0
1 + lim\frac{1}{x}⇢∞ \frac{sin\frac{1}{x}}{x} 1 + 0 = 0
Question 5: Solve, limx ⇢ π/2 (tanx)cosx
Solution:
let Y = limx ⇢ π/2 (tanx)cosx
Taking loge both sides,
logeY = limx ⇢ π/2 loge(tanx)cosx
logeY = limx ⇢ π/2 cosx loge(tanx)
logey = limx ⇢ π/2 loge(tanx)/secx
Using l-hospital,
logey = limx ⇢ π/2 cosx /sin2x = 0
Now, taking exponent on both sides,
Y = limx ⇢ π/2 e0
Y = limx ⇢ π/2 (tanx)cosx = 1
Question 6: limx ⇢ 0
Solution:
limx⇢0 \frac{1+\frac{x}{1!} + \frac{x2}{2!} + \frac{x3}{3!} – ( 1+ x+ \frac{x2}{2!} ) }{x3}
limx⇢0
\frac{\frac{x3}{3!}}{x3} = 1/3! =1/6
Question 7: Solve, lima ⇢ 0
Solution:
Using l-hospital (Differentiating numerator and denominator w.r.t a)
lima ⇢ 0 xalogx = logx
Question 8: Solve, limx ⇢ 0
Solution:
limx ⇢ 0
\frac{x^2+x-(x-x^3/3!)}{x^2} limx ⇢ 0 1 + x/3! = 1
Limit Formula
Limits help us comprehend how functions behave as their inputs approach certain values. Think of a limit as the destination that a function aims to reach as the input gets closer and closer to a specific point.
In this article, we will explore the essential limit formulas that form the backbone of calculus. These formulas are like the rules of a game, guiding us on how to find limits in various scenarios. Whether you’re adding, subtracting, multiplying, or dividing functions, there are specific formulas to help you determine the limit.
Table of Content
- What is a Limit in Mathematics?
- Limit Formulas
- Basic Limit Formulas
- Trigonometric Limits
- L-hospital Rule
- Exponential Limits
- Logarithmic limits
- Important Limit Results
- Sample Problem
- Practice Problems
- FAQs
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