my book defines a weak tangent as one where the line through [itex]\alpha(t_0 + h) [/itex] and [itex] \alpha(t_0) [/itex] has a limit position when [itex] h \rightarrow 0 [/itex]. they define a strong tangent as one where the line through [itex]\alpha(t_0 + h) [/itex] and [itex] \alpha(t_0 + k) [/itex] has a limit position when [itex] h, k \rightarrow 0 [/itex].(adsbygoogle = window.adsbygoogle || []).push({});

i am trying to show that the curve [itex] \alpha(t) = (t^3, t^2) [/itex] has a weak tangent at t = 0 but no strong tangent there.

i have that [itex] \alpha(0) = (0, 0) [/itex] and [itex] \alpha(h) = (h^3, h^2) [/itex]. i then have that [itex] \alpha(h) - \alpha(0) = (h^3, h^2) [/itex] and that [itex] \lim_{h \to 0} \frac{1}{h} (h^3, h^2) = \lim_{h \to 0} (h^2, h) = (0, 0) [/itex]. so a weak tangent exists.

analogously i try for the strong tangent with: [itex] \alpha(h) = (h^3, h^2) [/itex] and [itex] \alpha(k) = (k^3, k^2) [/itex] and [itex] \alpha(h) - \alpha(k) = (h^3 - k^3, h^2 - k^2) [/itex]. Then [itex] \lim_{h,k \to 0} \frac{1}{h-k} (h^3 - k^3, h^2 - k^2) = \lim_{h,k \to 0} (h^2 + hk + k^2, h + k) = (0, 0) [/itex].

but it seems like there is a strong tangent there since the limit exists. have i made a mistake somewhere?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Showing that a function does not have a strong tangent

Loading...

Similar Threads for Showing function does |
---|

I Approximating different functions |

I Equality between functions |

B How can I show the sum results in this? |

I Showing relationship between zeta and gamma |

B Wolfram Alpha graph of ln(x) shows as ln(abs(x)) |

**Physics Forums | Science Articles, Homework Help, Discussion**