Why does the inverse gamma distribution look essentially the same when increasing the variance?

I have a function in R which for a given mean $$\mu$$ and variance $$\sigma^2$$, spits out the parameters for the shape, $$\alpha$$, and rate, $$\beta$$ of the inverse gamma distribution with that mean and variance.

When setting the mean fixed at $$\mu=1$$, I noticed that increasing the variance from say 10 to 100, has almost no visual effect on the inverse gamma produced (both plots look identical to the figure below). Furthermore, when I sample from the corresponding distributions, the values don't seem to have a variance of higher than around 1 or 2. // mu=1, var=10 (alpha=2.1, beta=1.1)
> rinvgamma(50,2.1,1.1)
 0.9211008 0.4302407 0.5478425 0.5462700 0.3628111 1.1686150 1.0768032 1.2512244 1.7006653 0.7279801 0.5836686
 0.5233413 0.3448369 0.3536895 0.8559441 0.3650069 0.5783526 0.5317600 0.8353258 0.5587801 0.4564186 1.0995836
 0.3449476 0.5421546 0.1790850 7.3857559 0.6131200 0.6986830 0.2800351 0.5196334 0.3979346 0.4807825 0.3273610
 1.4138987 0.9715384 0.4951919 1.6518877 0.6431032 0.2710467 0.7001043 0.2214802 0.2868089 0.3429655 0.6041847
 0.2402038 0.3867785 0.2216013 0.9520140 0.5259651 0.5929160

// mu = 1, var=1000 (alpha=2.001, beta=1.001)
> rinvgamma(50,2.001,1.001)
 0.6629496 2.8095275 0.4344837 0.5361150 0.5480011 0.8444600 0.4915654 1.0122025 0.4696304 0.9146489 0.5594634
 0.2602492 0.6842792 0.6851688 0.4322968 3.0454766 0.2657547 0.2271760 0.7609194 0.1481709 0.9178390 0.3807237
 0.7069672 1.3271837 0.4251466 1.1065251 0.5227289 0.6202342 1.6205382 1.1839423 0.6605649 0.9626064 0.9334211
 0.6991242 0.7422693 0.2975816 1.0126841 0.4243968 0.6060229 0.3395812 2.5289663 4.1525575 0.8321886 1.4190376
 0.2293600 0.3870373 0.7642754 0.3934958 0.7532922 0.6021346

The $$\alpha$$ and $$\beta$$ values produced from the function ($$\alpha=2.1 \ , \beta=1.1$$ and $$\alpha=2.001 \ , \beta=1.001$$) seem to give the correct $$\mu$$ and $$\sigma^2$$ according to the mean-variance equations (see Wikipedia page here) so I'm puzzled why the distributions seem very similar.

Interestingly, if I increase the mean to $$\mu=100$$, the same effect is observed, but for larger variances (i.e. distributions with $$\mu=100$$ and $$\sigma^2=1000$$ vs. $$\sigma^2=2000$$ appear the same) - as if it's parameter combinations with a large relative difference between mean and variance values that are causing the issue.

Can anyone explain why I'm observing this effect?