What is the limit as x approaches infinity of (x^2+4)/(x^2+2) ?

The limit as x approaches infinity of (x^2+4)/(x^2+2) is 1

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limit as x approaches infinity of (x^2+4)/(x^2+2)

x \to \infty lim (\frac{x ^{2} + 4}{x ^{2} + 2})

1

Solution steps

x \to \infty lim (\frac{x ^{2} + 4}{x ^{2} + 2})

Divide by highest denominator power:

\frac{1 + \frac{4}{x ^{2}}}{1 + \frac{2}{x ^{2}}}

= x \to \infty lim (\frac{1 + \frac{4}{x ^{2}}}{1 + \frac{2}{x ^{2}}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( 1 + \frac{4}{x ^{2}} )}{lim _{x \to \infty} ( 1 + \frac{2}{x ^{2}} )}

x \to \infty lim (1 + \frac{4}{x ^{2}}) = 1

x \to \infty lim (1 + \frac{2}{x ^{2}}) = 1

= \frac{1}{1}

Simplify

= 1

x \to \infty lim ((1 + a x)^{\frac{1}{x}})

x \to - 1 lim (∣ x - 1 ∣ (x + 1))

x \to 5 - lim (\frac{x ^{2} + 7}{x ^{2} - 25})

x \to 0 lim ((e^{x} - x)^{\frac{3}{x}})

x \to - \frac{π}{2} lim (- 7 x - \frac{π}{2})

What is the limit as x approaches infinity of (x^2+4)/(x^2+2) ?
The limit as x approaches infinity of (x^2+4)/(x^2+2) is 1