Hunter

**Solution:**

**Concept:**The value is the value of the function such that the variable *x* approaches *a*. This is known as the limiting value of the function at . For limit to exist, the value of function when x approaches a from left is equal to the value of function when x approaches a from right, which is,

The value of the function at may vary from its limiting value at .

Few of the properties of limits are listed as shown below:

1.

2.

3.

4.

5.

There are also some important limits listed as below:

1.

2.

3.

4.

In the cases where any indeterminate form is obtained, the L-Hospital rule is applied which is the differentiation of numerator and denominator separately until the indeterminate form is removed.

**Calculation:**

**Step:1 **The function is and the limit *x* approaches to then it is written as, .

**Step:2 **Rationalise the expression by multiplying and dividing by .

**Step:3 **The value is calculated as shown below by dividing the numerator and denominator by *x*.

**Step:4 **Substitute *x* as in the expression and simplify as follows:

Thus, the limit value of is .

**Answer:**The limit value of is .

** **

** **

**Similar Problems:**

**Problem 1:**What is the limit as *x* approaches infinity of ?

**Solution:**

**Step:1 **The function is and the limit *x* approaches to then it is written as, .

**Step:2 **The tangent function has value between and for every interval therefore at infinity, it may have any value in between.

**Step:3 **Therefore, the value of for is not defined.

Thus, the limit does not exist.

**Answer:** The limit does not exist.

** **

**Problem 2:**What is the limit as *x* approaches infinity of ?

**Solution:**

**Step:1 **The function is and the limit *x* approaches to then it is written as, .

**Step:2 **Since , use this result and simplify the value of limit as shown below:

**Step:3 **Replace by *t* in the expression and then simplify.

Thus, the limit value of is 1.

**Answer:** The limit value of is 1.

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