Limits MCQ Quiz in తెలుగు - Objective Question with Answer for Limits - ముఫ్త్ [PDF] డౌన్‌లోడ్ కరెన్

Concept:

Limit:

The number L is called the limit of function f(x) as x→a if and only if, for every ε > 0 there exists δ > 0 such that:

|f(x) - L|

Whenever 0

Calculation:

Given that:

xn = 0.5xn−1 + 1 

x₁ = 1 (x₀ = 0 given)

The sequence is:

Explanation:

substitue x = π

, sin π = 0

We can see that denominator becomes zero.

Thus for p to have finite value numerator is also zero.

π² + απ + 2π² = 0

α = - 3π

Now,  form

Using L'Hospital's rule, we get

substituting x = π, α = - 3π

, cosπ = -1

p = 

p = π 

Concept:

L’ Hospital’s Rule:

This rule is only applicable for  and  indeterminate forms.

Calculation

As 

Which is an indeterminate form.

Using L hospitality rule for 

We get

Again applying the L hospitality rule, we get 

∴ Value of 

Taking log_e on both sides

Substituting x = ∞

It is of the form  

Differentiating w.r.t x

Substituting x = ∞

∴ y = e⁰

∴ y = 1

Not defined 

Using L – Hospital’s rule,

1^st derivation

Not defined

2^nd derivation

Not defined

3^rd derivation

Concept: 

L-Hospital Rule: Let f(x) and g(x) be two functions

Suppose that we have one of the following cases,

I.  

II. 

Then we can apply L-Hospital Rule as:

Calculation:

Given​​

Applying L’ Hospital  rule:

Once again by L’ Hospital rule,

Concept:

Calculation:

Given:

cos² x = 1 – sin² x

Multiply and divide by π sin²x

, the above expression gives:

= π = 3.14

Concept:

L’ Hospital’s Rule:

This rule is only applicable for  and  indeterminate forms.

Explanation:

 which is in the form of 0/0

So, applying L- Hospital rule twice,

then,

= 

∴ For the limit to be finite, the x² term in the numerator should be 0.

Now,

Concept:

A function f(x) is said to be continuous at a point x = a, in its domain if,

 exists or its graph is a single unbroken curve.

f(x) is Continuous at x = a

Calculation:

Given:

Left hand limit and right hand limit are not equal 

∴ does not exist.

∴ All given options are true.