lim x->∞ pow top = pow bottom
coefficients
To evaluate the limit as x approaches infinity for the expression, we need to consider the highest power of x in the numerator and denominator. If the highest powers of x in the numerator and denominator are the same, then we can take the ratio of the coefficients of the highest power to find the limit.
For example, suppose we have the expression:
lim x->∞ (5x^3 + 2x^2 – x)/(3x^3 – 5x^2 + 4x)
In this expression, the highest power of x in both the numerator and denominator is 3. Therefore, we take the ratio of the coefficients of the highest power to find the limit, and simplify.
lim x->∞ (5x^3 + 2x^2 – x)/(3x^3 – 5x^2 + 4x)
= lim x->∞ (5 + 2/x – 1/x^2)/(3 – 5/x + 4/x^2)
= 5/3
Therefore, the limit as x approaches infinity for the given expression is 5/3.
If the highest powers of x in the numerator and denominator are not the same, then we need to divide both numerator and denominator by the highest power of x and simplify. This will help us to take the limit.
For example, suppose we have the expression:
lim x->∞ (2x^3 + 3x^2 – x)/(5x^4 + 2x^3 – 7x)
In this expression, the highest power of x in the numerator is 3 and in the denominator is 4. To simplify this, we can divide both numerator and denominator by x^4.
lim x->∞ (2x^3/x^4 + 3x^2/x^4 – x/x^4)/(5x^4/x^4 + 2x^3/x^4 – 7x/x^4)
= lim x->∞ (2/x + 3/x^2 – 1/x^3)/(5 + 2/x – 7/x^3)
= 0/5
= 0
Therefore, the limit as x approaches infinity for the given expression is 0.
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