**Evaluating Limits at Infinity :**

Here we are going to see how to evaluate limits at infinity.

By comparing the degree of the given rational expression, we may decide the answer. Here you may find some shortcuts.

Degree of the numerator is less than the denominator |
lim Answer will be 0. |

Degree of the numerator is same as the denominator |
lim (3x Answer will be 3/7. |

Degree of the numerator is greater than the denominator |
lim (x Answer is does not exists. |

Now let us look into some example problems on evaluating limits at infinity.

**Question 1 :**

lim _{x->}_{∞} (x^{3} + x)/(x^{4} - 3x^{2} + 1)

**Solution :**

**f(x) = **(x^{3} + x)/(x^{4} - 3x^{2} + 1)

Divide each terms by x^{4}, we get

**f(x) = **(1/x + 1/x^{3})/(1 - 3/x^{2} + 1/x^{4})

lim _{x->}_{∞} (x^{3} + x)/(x^{4} - 3x^{2} + 1)

= lim _{x->}_{∞} (1/x + 1/x^{3})/(1 - 3/x^{2} + 1/x^{4})

By applying the limit, we get

= 0

Hence the value of lim _{x->}_{∞} (x^{3} + x)/(x^{4} - 3x^{2} + 1) is 0.

**Question 2 :**

lim _{x->}_{∞} (x^{4} - 5x)/(x^{2} - 3x + 1)

**Solution :**

**f(x) = **(x^{4} - 5x)/(x^{2} - 3x + 1)

By dividing the highest exponent of denominator, we get

**f(x) = **(x^{2} - 5/x)/(1 - 3/x + 1/x^{2})

= lim _{x->}_{∞}(x^{2} - 5/x)/(1 - 3/x + 1/x^{2})

By applying the limit value, we get

= ∞

Hence the value of lim _{x->}_{∞} (x^{4} - 5x)/(x^{2} - 3x + 1) is ∞.

**Question 3 :**

lim _{x->}_{∞} [(1 + x - 3x^{3})/(1 + x^{2} + 3x^{3})]

**Solution :**

**f(x) = ** [(1 + x - 3x^{3})/(1 + x^{2} + 3x^{3})]

**f(x) = ** [(1/x^{3} + 1/x^{2} - 3)/(1/x^{3} + 1/x + 3)]

**f(x) = ** lim _{x->}_{∞ }[(1/x^{3} + 1/x^{2} - 3)/(1/x^{3} + 1/x + 3)]

By applying the limit ∞ in the question, we get

= -3/3

= -1

Hence the value of lim _{x->}_{∞} [(1 + x - 3x^{3})/(1 + x^{2} + 3x^{3})] is -1.

**Question 4 :**

lim _{x->}_{∞} [x^{3}/(2x^{2} - 1) - x^{2}/(2x + 1)]

**Solution :**

**f(x) = **[x^{3}/(2x^{2} - 1) - x^{2}/(2x + 1)]

** = **[x^{3}(2x + 1) - x^{2}(2x^{2} - 1)] / (2x + 1)(2x^{2} - 1)

** = **[2x^{4 }+ x^{3} - 2x^{4} + x^{2}] / (4x^{3} - 2x + 2x^{2} - 1)

** = (**x^{3} + x^{2}) / (4x^{3} - 2x + 2x^{2} - 1)

Divide by x^{3,} we get

** = (1** + (1/x)) / (4 - 2/x^{2} + 2/x - 1/x^{3})

By applying the limit value, we get

= 1/4

Hence the value of lim _{x->}_{∞} [x^{3}/(2x^{2} - 1) - x^{2}/(2x + 1)] is 1/4.

After having gone through the stuff given above, we hope that the students would have understood, "Evaluating Limits at Infinity"

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