The limit of tanx/x as x approaches infinity does not exist, that is, lim_{x→∞} tanx/x is undefined. Here we learn to find the limit of tanx/x when x tends to infinity.

The formula of the limit of tanx/x when x approaches ∞ is given by lim_{x→∞} tanx/x = undefined. Below theorem will be useful to prove this formula.

**Theorem:** If lim_{x→c} f(x) exists, then for every sequence {x_{n}} converging to c the limit lim_{n→∞} f(x_{n}) is always unique.

## What is the Limit of tanx/x when x→∞

**Answer:** The value of the limit of tanx/x when x→∞ is undefined.

*Proof:*

Let f(x) = $\dfrac{\tan x}{x}$.

In order to show lim_{x→∞} f(x) does not exist, we consider two sequences {x_{n}} and {y_{n}}, both converging to ∞, but f(x_{n}) and f(y_{n}) converge to different limits. Let us assume that

x_{n} = nπ

y_{n} = π/2 + nπ – 1/n^{2}

Here, both x_{n}, y_{n} →∞, but we have that

lim_{n→∞} f(x_{n}) = lim_{n→∞} $\frac{\tan n\pi}{n \pi}$ = 0.

On the other hand,

lim_{n→∞} f(y_{n}) = lim_{n→∞} $\frac{\tan (\pi/2 + n\pi-1/n^2)}{\pi/2 + n\pi-1/n^2}$ = ∞.

As both sequences f(x_{n}) and f(y_{n}) converge to different limits, by the above theorem we conclude the following:

The limit of tanx/x when x→∞ does not exist.

**ALSO READ:**

lim_{x→0} (e^{x}-1)/x = 1 | lim_{x→∞} sinx/x = 0 |

lim_{x→0} x/cosx = 0 | lim_{x→0} x/sinx = 1 |

lim_{x→0} tanx/x = 1 | Sum rule of limits |

Product rule of limits | Quotient rule of limits |

ε – δ definition of limit |

## FAQs

### Q1: Find the limit of tanx/x as x tends to ∞.

Answer: The limit of tanx/x as x tends to ∞ is not defined.